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<b>let </b><font color="Maroon" title="c1">g0</font>, <font color="Maroon" title="c2">g1</font>, <font color="Maroon" title="c3">g2</font>, <font color="Maroon" title="c4">g3</font>, <font color="Maroon" title="c5">g4</font>, <font color="Maroon" title="c6">g5</font>, <font color="Maroon" title="c7">g6</font>, <font color="Maroon" title="c8">g7</font>, <font color="Maroon" title="c9">g8</font>, <font color="Maroon" title="c10">g9</font>, <font color="Maroon" title="c11">g10</font>, <font color="Maroon" title="c12">g11</font>, <font color="Maroon" title="c13">g12</font>, <font color="Maroon" title="c14">g13</font>, <font color="Maroon" title="c15">g14</font>, <font color="Maroon" title="c16">g15</font>, <font color="Maroon" title="c17">g16</font>, <font color="Maroon" title="c18">a0</font>, <font color="Maroon" title="c19">a1</font>, <font color="Maroon" title="c20">a2</font>, <font color="Maroon" title="c21">a3</font>, <font color="Maroon" title="c22">a4</font>, <font color="Maroon" title="c23">a5</font>, <font color="Maroon" title="c24">a6</font>, <font color="Maroon" title="c25">a7</font>, <font color="Maroon" title="c26">a8</font>, <font color="Maroon" title="c27">a9</font>, <font color="Maroon" title="c28">a10</font>, <font color="Maroon" title="c29">a11</font>, <font color="Maroon" title="c30">a12</font>, <font color="Maroon" title="c31">a13</font>, <font color="Maroon" title="c32">a14</font>, <font color="Maroon" title="c33">a15</font>, <font color="Maroon" title="c34">b0</font>, <font color="Maroon" title="c35">b1</font>, <font color="Maroon" title="c36">b2</font>, <font color="Maroon" title="c37">b3</font>, <font color="Maroon" title="c38">b4</font>, <font color="Maroon" title="c39">b5</font>, <font color="Maroon" title="c40">b6</font>, <font color="Maroon" title="c41">b7</font>, <font color="Maroon" title="c42">b8</font>, <font color="Maroon" title="c43">b9</font>, <font color="Maroon" title="c44">b10</font>, <font color="Maroon" title="c45">b11</font>, <font color="Maroon" title="c46">b12</font>, <font color="Maroon" title="c47">b13</font>, <font color="Maroon" title="c48">b14</font>, <font color="Maroon" title="c49">b15</font>, <font color="Maroon" title="c50">z</font>, <font color="Maroon" title="c51">p</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"> (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">g0</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c17">g16</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c50">z</font> &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c51">p</font>,<font color="Maroon" title="c33">a15</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c51">p</font>,<font color="Maroon" title="c33">a15</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c34">b0</font></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c34">b0</font></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c34">b0</font></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c34">b0</font></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c34">b0</font></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c34">b0</font></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c34">b0</font></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c34">b0</font></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c34">b0</font></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c34">b0</font></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c34">b0</font></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c34">b0</font></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c34">b0</font></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c34">b0</font></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c42">b8</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c34">b0</font></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c34">b0</font></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c42">b8</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c43">b9</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c34">b0</font></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c34">b0</font></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c43">b9</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c44">b10</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c34">b0</font></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c34">b0</font></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c44">b10</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c45">b11</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c34">b0</font></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c34">b0</font></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c45">b11</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c46">b12</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">a11</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c34">b0</font></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">a11</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c34">b0</font></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c46">b12</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c47">b13</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c30">a12</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c34">b0</font></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c30">a12</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c34">b0</font></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c47">b13</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c48">b14</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c31">a13</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c34">b0</font></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c31">a13</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c34">b0</font></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c48">b14</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c49">b15</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c32">a14</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c34">b0</font></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c32">a14</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c34">b0</font></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c49">b15</font> ) implies ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c49">b15</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c32">a14</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c32">a14</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c49">b15</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c48">b14</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c31">a13</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c31">a13</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c48">b14</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c47">b13</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c30">a12</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c30">a12</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c47">b13</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c46">b12</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">a11</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">a11</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c46">b12</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c45">b11</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c45">b11</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c44">b10</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c44">b10</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c43">b9</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c43">b9</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c42">b8</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c42">b8</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> ) ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:4"/><i><font color="Green" title="E1">A1</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">g0</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c17">g16</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c50">z</font> )
 <b>and </b><br/><a NAME="E2:4"/><i><font color="Green" title="E2">A2</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c51">p</font>,<font color="Maroon" title="c33">a15</font> )
 <b>and </b><br/><a NAME="E3:4"/><i><font color="Green" title="E3">A3</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c34">b0</font></span>)</span> )
 <b>and </b><br/><a NAME="E4:4"/><i><font color="Green" title="E4">A4</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c34">b0</font></span>)</span> )
 <b>and </b><br/><a NAME="E5:4"/><i><font color="Green" title="E5">A5</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c34">b0</font></span>)</span> )
 <b>and </b><br/><a NAME="E6:4"/><i><font color="Green" title="E6">A6</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c34">b0</font></span>)</span> )
 <b>and </b><br/><a NAME="E7:4"/><i><font color="Green" title="E7">A7</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c34">b0</font></span>)</span> )
 <b>and </b><br/><a NAME="E8:4"/><i><font color="Green" title="E8">A8</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c34">b0</font></span>)</span> )
 <b>and </b><br/><a NAME="E9:4"/><i><font color="Green" title="E9">A9</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c34">b0</font></span>)</span> )
 <b>and </b><br/><a NAME="E10:4"/><i><font color="Green" title="E10">A10</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c42">b8</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c34">b0</font></span>)</span> )
 <b>and </b><br/><a NAME="E11:4"/><i><font color="Green" title="E11">A11</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c43">b9</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c34">b0</font></span>)</span> )
 <b>and </b><br/><a NAME="E12:4"/><i><font color="Green" title="E12">A12</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c44">b10</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c34">b0</font></span>)</span> )
 <b>and </b><br/><a NAME="E13:4"/><i><font color="Green" title="E13">A13</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c45">b11</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c34">b0</font></span>)</span> )
 <b>and </b><br/><a NAME="E14:4"/><i><font color="Green" title="E14">A14</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c46">b12</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">a11</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c34">b0</font></span>)</span> )
 <b>and </b><br/><a NAME="E15:4"/><i><font color="Green" title="E15">A15</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c47">b13</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c30">a12</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c34">b0</font></span>)</span> )
 <b>and </b><br/><a NAME="E16:4"/><i><font color="Green" title="E16">A16</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c48">b14</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c31">a13</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c34">b0</font></span>)</span> )
 <b>and </b><br/><a NAME="E17:4"/><i><font color="Green" title="E17">A17</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c49">b15</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c32">a14</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c34">b0</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c49">b15</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c32">a14</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c32">a14</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c49">b15</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c48">b14</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c31">a13</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c31">a13</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c48">b14</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c47">b13</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c30">a12</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c30">a12</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c47">b13</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c46">b12</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">a11</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">a11</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c46">b12</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c45">b11</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c45">b11</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c44">b10</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c44">b10</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c43">b9</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c43">b9</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c42">b8</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c42">b8</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> ) )</span><br/>

<a NAME="E18:4"/>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c32">a14</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c32">a14</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) ) 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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c32">a14</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E19:4"/><b>then </b>
( ( ( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c32">a14</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c33">a15</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c32">a14</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c33">a15</font> ) ) &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c51">p</font> ) or ( not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c32">a14</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c33">a15</font> ) &amp; not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c32">a14</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c32">a14</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E1:4"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E20:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c32">a14</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c16">g15</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c16">g15</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c32">a14</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c16">g15</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c16">g15</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c32">a14</font> or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c16">g15</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c32">a14</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c16">g15</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c16">g15</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c32">a14</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>;<br/>
<a NAME="E21:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c32">a14</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c34">b0</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c32">a14</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c34">b0</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c32">a14</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E2:4"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E22:4"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c49">b15</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c32">a14</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c16">g15</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="gate_4.html#E17:4"><i><font color="Green" title="E17">A17</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">$</a> <font color="Maroon" title="c48">b14</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c31">a13</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c31">a13</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c48">b14</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c47">b13</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c30">a12</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c30">a12</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c47">b13</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c46">b12</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">a11</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">a11</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c46">b12</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c45">b11</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c45">b11</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c44">b10</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c44">b10</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c43">b9</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c43">b9</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c42">b8</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c42">b8</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> ) )</span><br/>

<a NAME="E23:4"/>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c31">a13</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c31">a13</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) ) 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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c31">a13</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E24:4"/><b>then </b>
( ( ( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c31">a13</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c33">a15</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c31">a13</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c33">a15</font> ) ) &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c51">p</font> ) or ( not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c31">a13</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c33">a15</font> ) &amp; not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c31">a13</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c31">a13</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E1:4"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E25:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c31">a13</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">g14</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">g14</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c31">a13</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">g14</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">g14</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c31">a13</font> or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">g14</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c31">a13</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">g14</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">g14</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c31">a13</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>;<br/>
<a NAME="E26:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c31">a13</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c34">b0</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c31">a13</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c34">b0</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c31">a13</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E2:4"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E27:4"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c48">b14</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c31">a13</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c15">g14</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="gate_4.html#E16:4"><i><font color="Green" title="E16">A16</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">$</a> <font color="Maroon" title="c47">b13</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c30">a12</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c30">a12</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c47">b13</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c46">b12</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">a11</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">a11</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c46">b12</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c45">b11</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c45">b11</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c44">b10</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c44">b10</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c43">b9</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c43">b9</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c42">b8</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c42">b8</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> ) )</span><br/>

<a NAME="E28:4"/>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c30">a12</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c30">a12</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) ) 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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c30">a12</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E29:4"/><b>then </b>
( ( ( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c30">a12</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c33">a15</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c30">a12</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c33">a15</font> ) ) &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c51">p</font> ) or ( not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c30">a12</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c33">a15</font> ) &amp; not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c30">a12</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c30">a12</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E1:4"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E30:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c30">a12</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c14">g13</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c14">g13</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c30">a12</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c14">g13</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c14">g13</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c30">a12</font> or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c14">g13</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c30">a12</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c14">g13</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c14">g13</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c30">a12</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>;<br/>
<a NAME="E31:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c30">a12</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c34">b0</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c30">a12</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c34">b0</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c30">a12</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E2:4"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E32:4"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c47">b13</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c30">a12</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">g13</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="gate_4.html#E15:4"><i><font color="Green" title="E15">A15</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">$</a> <font color="Maroon" title="c46">b12</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">a11</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">a11</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c46">b12</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c45">b11</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c45">b11</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c44">b10</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c44">b10</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c43">b9</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c43">b9</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c42">b8</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c42">b8</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> ) )</span><br/>

<a NAME="E33:4"/>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">a11</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">a11</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) ) 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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">a11</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E34:4"/><b>then </b>
( ( ( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c29">a11</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c33">a15</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c29">a11</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c33">a15</font> ) ) &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c51">p</font> ) or ( not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c29">a11</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c33">a15</font> ) &amp; not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c29">a11</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">a11</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E1:4"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E35:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c29">a11</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c13">g12</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c13">g12</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c29">a11</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c13">g12</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c13">g12</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c29">a11</font> or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c13">g12</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c29">a11</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c13">g12</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c13">g12</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">a11</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>;<br/>
<a NAME="E36:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c29">a11</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c34">b0</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c29">a11</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c34">b0</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">a11</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E2:4"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E37:4"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c46">b12</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">a11</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c13">g12</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="gate_4.html#E14:4"><i><font color="Green" title="E14">A14</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">$</a> <font color="Maroon" title="c45">b11</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c45">b11</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c44">b10</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c44">b10</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c43">b9</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c43">b9</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c42">b8</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c42">b8</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> ) )</span><br/>

<a NAME="E38:4"/>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) ) 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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E39:4"/><b>then </b>
( ( ( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c28">a10</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c28">a10</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font> ) ) &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c51">p</font> ) or ( not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c28">a10</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font> ) &amp; not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c28">a10</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E1:4"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E40:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c28">a10</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">g11</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">g11</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c28">a10</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">g11</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">g11</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c28">a10</font> or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">g11</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c28">a10</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">g11</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">g11</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>;<br/>
<a NAME="E41:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c28">a10</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c34">b0</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c28">a10</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c34">b0</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E2:4"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E42:4"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c45">b11</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c28">a10</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">g11</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="gate_4.html#E13:4"><i><font color="Green" title="E13">A13</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">$</a> <font color="Maroon" title="c44">b10</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c44">b10</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c43">b9</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c43">b9</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c42">b8</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c42">b8</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> ) )</span><br/>

<a NAME="E43:4"/>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) ) 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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E44:4"/><b>then </b>
( ( ( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c27">a9</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c27">a9</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font> ) ) &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font> ) or ( not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c27">a9</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font> ) &amp; not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c27">a9</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E1:4"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E45:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c27">a9</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">g10</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">g10</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c27">a9</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">g10</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">g10</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c27">a9</font> or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">g10</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c27">a9</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">g10</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">g10</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>;<br/>
<a NAME="E46:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c27">a9</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c34">b0</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c27">a9</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c34">b0</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E2:4"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E47:4"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c44">b10</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c27">a9</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">g10</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="gate_4.html#E12:4"><i><font color="Green" title="E12">A12</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">$</a> <font color="Maroon" title="c43">b9</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c43">b9</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c42">b8</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c42">b8</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> ) )</span><br/>

<a NAME="E48:4"/>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) ) 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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E49:4"/><b>then </b>
( ( ( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c26">a8</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c26">a8</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font> ) ) &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font> ) or ( not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c26">a8</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font> ) &amp; not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c26">a8</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E1:4"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E50:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c26">a8</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">g9</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">g9</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c26">a8</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">g9</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">g9</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c26">a8</font> or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">g9</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c26">a8</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">g9</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">g9</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>;<br/>
<a NAME="E51:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c26">a8</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c34">b0</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c26">a8</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c34">b0</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E2:4"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E52:4"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c43">b9</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">a8</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">g9</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="gate_4.html#E11:4"><i><font color="Green" title="E11">A11</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">$</a> <font color="Maroon" title="c42">b8</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c42">b8</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> ) )</span><br/>

<a NAME="E53:4"/>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) ) 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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E54:4"/><b>then </b>
( ( ( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c25">a7</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c25">a7</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font> ) ) &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font> ) or ( not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c25">a7</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font> ) &amp; not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c25">a7</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E1:4"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E55:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c25">a7</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">g8</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">g8</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c25">a7</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">g8</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">g8</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c25">a7</font> or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">g8</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c25">a7</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">g8</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">g8</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>;<br/>
<a NAME="E56:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c25">a7</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c34">b0</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c25">a7</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c34">b0</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E2:4"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E57:4"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c42">b8</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c25">a7</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">g8</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="gate_4.html#E10:4"><i><font color="Green" title="E10">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">$</a> <font color="Maroon" title="c41">b7</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> ) )</span><br/>

<a NAME="E58:4"/>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) ) 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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E59:4"/><b>then </b>
( ( ( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c24">a6</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c24">a6</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font> ) ) &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font> ) or ( not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c24">a6</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font> ) &amp; not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c24">a6</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E1:4"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E60:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c24">a6</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">g7</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">g7</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c24">a6</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">g7</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">g7</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c24">a6</font> or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">g7</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c24">a6</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">g7</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">g7</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>;<br/>
<a NAME="E61:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c24">a6</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c34">b0</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c24">a6</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c34">b0</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E2:4"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E62:4"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">b7</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">a6</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c8">g7</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="gate_4.html#E9:4"><i><font color="Green" title="E9">A9</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">$</a> <font color="Maroon" title="c40">b6</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> ) )</span><br/>

<a NAME="E63:4"/>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) ) 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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E64:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c23">a5</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c23">a5</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font> ) or ( (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c23">a5</font> or  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c23">a5</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font> ) ) iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E1:4"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E65:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c23">a5</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">g6</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">g6</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c23">a5</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">g6</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">g6</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c23">a5</font> or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">g6</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c23">a5</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">g6</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">g6</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>;<br/>
<a NAME="E66:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c23">a5</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c34">b0</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c23">a5</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c34">b0</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E2:4"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E67:4"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">b6</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c23">a5</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">g6</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="gate_4.html#E8:4"><i><font color="Green" title="E8">A8</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">$</a> <font color="Maroon" title="c39">b5</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> ) )</span><br/>

<a NAME="E68:4"/>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) ) 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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E69:4"/><b>then </b>
( ( ( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c22">a4</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c22">a4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font> ) ) &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font> ) or ( not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c22">a4</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font> ) &amp; not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c22">a4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E1:4"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E70:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c22">a4</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">g5</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">g5</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c22">a4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">g5</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">g5</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c22">a4</font> or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">g5</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c22">a4</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">g5</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">g5</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>;<br/>
<a NAME="E71:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c22">a4</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c34">b0</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c22">a4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c34">b0</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E2:4"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E72:4"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">b5</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c22">a4</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">g5</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="gate_4.html#E7:4"><i><font color="Green" title="E7">A7</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">$</a> <font color="Maroon" title="c38">b4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> ) )</span><br/>

<a NAME="E73:4"/>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) ) 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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E74:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">a3</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">a3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font> ) or ( (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">a3</font> or  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">a3</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font> ) ) iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E1:4"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E75:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">a3</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">g4</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">g4</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">a3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">g4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">g4</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">a3</font> or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">g4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">a3</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">g4</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">g4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>;<br/>
<a NAME="E76:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">a3</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c34">b0</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">a3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c34">b0</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E2:4"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E77:4"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">b4</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c21">a3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">g4</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="gate_4.html#E6:4"><i><font color="Green" title="E6">A6</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">$</a> <font color="Maroon" title="c37">b3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> ) )</span><br/>

<a NAME="E78:4"/>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) ) 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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E79:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c20">a2</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c20">a2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font> ) or ( (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c20">a2</font> or  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c20">a2</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font> ) ) iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E1:4"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E80:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c20">a2</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">g3</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">g3</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c20">a2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">g3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">g3</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c20">a2</font> or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">g3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c20">a2</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">g3</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">g3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>;<br/>
<a NAME="E81:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c20">a2</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c34">b0</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c20">a2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c34">b0</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E2:4"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E82:4"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">b3</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c20">a2</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">g3</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="gate_4.html#E5:4"><i><font color="Green" title="E5">A5</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">$</a> <font color="Maroon" title="c36">b2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> ) )</span><br/>

<a NAME="E83:4"/>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) ) 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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E84:4"/><b>then </b>
( ( ( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c19">a1</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c19">a1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font> ) ) &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font> ) or ( not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c19">a1</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font> ) &amp; not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c19">a1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E1:4"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E85:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c19">a1</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">g2</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">g2</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c19">a1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">g2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">g2</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c19">a1</font> or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">g2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c19">a1</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">g2</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">g2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>;<br/>
<a NAME="E86:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c19">a1</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c34">b0</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c19">a1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c34">b0</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E2:4"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E87:4"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">b2</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">a1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">g2</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="gate_4.html#E4:4"><i><font color="Green" title="E4">A4</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">$</a> <font color="Maroon" title="c35">b1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> ) )</span><br/>

<a NAME="E88:4"/>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) ) 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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E89:4"/><b>then </b>
( ( ( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">a0</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">a0</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font> ) ) &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font> ) or ( not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">a0</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font> ) &amp; not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">a0</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E1:4"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E90:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">a0</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">g1</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">g1</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">a0</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">g1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">g1</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">a0</font> or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">g1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">a0</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">g1</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">g1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>;<br/>
<a NAME="E91:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">a0</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c34">b0</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">a0</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c34">b0</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E2:4"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E92:4"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">b1</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">a0</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">g1</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="gate_4.html#E3:4"><i><font color="Green" title="E3">A3</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">$</a> <font color="Maroon" title="c34">b0</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )</span><br/>

<a NAME="E93:4"/>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</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">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</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">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span> ) ) 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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E94:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c50">z</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c50">z</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font> ) or ( (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c50">z</font> or  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c50">z</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font> ) ) iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_4.html#E1:4"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<a NAME="E95:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c50">z</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">g0</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">g0</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c50">z</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">g0</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">g0</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</font> ) ) or ( (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c50">z</font> or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">g0</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c50">z</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">g0</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">a15</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">g0</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c51">p</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 
<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="E96:4"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">b0</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</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#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c33">a15</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c50">z</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">g0</font>,<font color="Maroon" title="c51">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="gate_4.html#E1:4"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="gate_4.html#E2:4"><i><font color="Green" title="E2">A2</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"> verum</span><br/>


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