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<span class="kw">let </span><font color="Maroon" title="c1">s0</font>, <font color="Maroon" title="c2">s1</font>, <font color="Maroon" title="c3">s2</font>, <font color="Maroon" title="c4">s3</font>, <font color="Maroon" title="c5">s4</font>, <font color="Maroon" title="c6">s5</font>, <font color="Maroon" title="c7">s6</font>, <font color="Maroon" title="c8">s7</font>, <font color="Maroon" title="c9">ns0</font>, <font color="Maroon" title="c10">ns1</font>, <font color="Maroon" title="c11">ns2</font>, <font color="Maroon" title="c12">ns3</font>, <font color="Maroon" title="c13">ns4</font>, <font color="Maroon" title="c14">ns5</font>, <font color="Maroon" title="c15">ns6</font>, <font color="Maroon" title="c16">ns7</font>, <font color="Maroon" title="c17">q1</font>, <font color="Maroon" title="c18">q2</font>, <font color="Maroon" title="c19">q3</font>, <font color="Maroon" title="c20">nq1</font>, <font color="Maroon" title="c21">nq2</font>, <font color="Maroon" title="c22">nq3</font>, <font color="Maroon" title="c23">R</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <!--HUMANRECFAILED-->( ( not <font color="Maroon" title="c1">s0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c19">q3</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c18">q2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c19">q3</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c18">q2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c1">s0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c2">s1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c19">q3</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c18">q2</font></span>)</span>,<font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c19">q3</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c18">q2</font></span>)</span>,<font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c2">s1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c3">s2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c19">q3</font></span>)</span>,<font color="Maroon" title="c18">q2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c19">q3</font></span>)</span>,<font color="Maroon" title="c18">q2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c3">s2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c4">s3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c19">q3</font></span>)</span>,<font color="Maroon" title="c18">q2</font>,<font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c19">q3</font></span>)</span>,<font color="Maroon" title="c18">q2</font>,<font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c4">s3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c5">s4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c19">q3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c18">q2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c19">q3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c18">q2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c5">s4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c6">s5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c19">q3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c18">q2</font></span>)</span>,<font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c19">q3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c18">q2</font></span>)</span>,<font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c6">s5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c7">s6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c19">q3</font>,<font color="Maroon" title="c18">q2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c19">q3</font>,<font color="Maroon" title="c18">q2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c7">s6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c8">s7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c19">q3</font>,<font color="Maroon" title="c18">q2</font>,<font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c19">q3</font>,<font color="Maroon" title="c18">q2</font>,<font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c8">s7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c22">nq3</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c21">nq2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c20">nq1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c22">nq3</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c21">nq2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c20">nq1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c10">ns1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c22">nq3</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c21">nq2</font></span>)</span>,<font color="Maroon" title="c20">nq1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c22">nq3</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c21">nq2</font></span>)</span>,<font color="Maroon" title="c20">nq1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c10">ns1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c11">ns2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c22">nq3</font></span>)</span>,<font color="Maroon" title="c21">nq2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c20">nq1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c22">nq3</font></span>)</span>,<font color="Maroon" title="c21">nq2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c20">nq1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c11">ns2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c12">ns3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c22">nq3</font></span>)</span>,<font color="Maroon" title="c21">nq2</font>,<font color="Maroon" title="c20">nq1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c22">nq3</font></span>)</span>,<font color="Maroon" title="c21">nq2</font>,<font color="Maroon" title="c20">nq1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c12">ns3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c13">ns4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c22">nq3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c21">nq2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c20">nq1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c22">nq3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c21">nq2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c20">nq1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c13">ns4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c14">ns5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c22">nq3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c21">nq2</font></span>)</span>,<font color="Maroon" title="c20">nq1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c22">nq3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c21">nq2</font></span>)</span>,<font color="Maroon" title="c20">nq1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c14">ns5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c22">nq3</font>,<font color="Maroon" title="c21">nq2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c20">nq1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c22">nq3</font>,<font color="Maroon" title="c21">nq2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c20">nq1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c22">nq3</font>,<font color="Maroon" title="c21">nq2</font>,<font color="Maroon" title="c20">nq1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c22">nq3</font>,<font color="Maroon" title="c21">nq2</font>,<font color="Maroon" title="c20">nq1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c20">nq1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c20">nq1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c21">nq2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c17">q1</font>,<font color="Maroon" title="c18">q2</font></span>)</span>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c17">q1</font>,<font color="Maroon" title="c18">q2</font></span>)</span>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c21">nq2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c22">nq3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c19">q3</font>,<span class="p3">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c17">q1</font>,<span class="p3">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">q2</font>,<font color="Maroon" title="c19">q3</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c19">q3</font>,<span class="p3">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c17">q1</font>,<span class="p3">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">q2</font>,<font color="Maroon" title="c19">q3</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c22">nq3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; not <!--HUMANRECFAILED-->( ( not <font color="Maroon" title="c10">ns1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">s0</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">s0</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c10">ns1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c11">ns2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c11">ns2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c12">ns3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">s2</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">s2</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c12">ns3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c13">ns4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">s3</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">s3</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c13">ns4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c14">ns5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">s4</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">s4</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c14">ns5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">s5</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">s5</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">s6</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">s6</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c8">s7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c23">R</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c8">s7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c23">R</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:4"/><span class="lab"><font color="Green" title="E1">A1</font></span>: 
( not <font color="Maroon" title="c1">s0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c19">q3</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c18">q2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E2:4"/><span class="lab"><font color="Green" title="E2">A2</font></span>: 
( not <font color="Maroon" title="c2">s1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c19">q3</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c18">q2</font></span>)</span>,<font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E3:4"/><span class="lab"><font color="Green" title="E3">A3</font></span>: 
( not <font color="Maroon" title="c3">s2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c19">q3</font></span>)</span>,<font color="Maroon" title="c18">q2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E4:4"/><span class="lab"><font color="Green" title="E4">A4</font></span>: 
( not <font color="Maroon" title="c4">s3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c19">q3</font></span>)</span>,<font color="Maroon" title="c18">q2</font>,<font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E5:4"/><span class="lab"><font color="Green" title="E5">A5</font></span>: 
( not <font color="Maroon" title="c5">s4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c19">q3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c18">q2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E6:4"/><span class="lab"><font color="Green" title="E6">A6</font></span>: 
( not <font color="Maroon" title="c6">s5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c19">q3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c18">q2</font></span>)</span>,<font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E7:4"/><span class="lab"><font color="Green" title="E7">A7</font></span>: 
( not <font color="Maroon" title="c7">s6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c19">q3</font>,<font color="Maroon" title="c18">q2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E8:4"/><span class="lab"><font color="Green" title="E8">A8</font></span>: 
( not <font color="Maroon" title="c8">s7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c19">q3</font>,<font color="Maroon" title="c18">q2</font>,<font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E9:4"/><span class="lab"><font color="Green" title="E9">A9</font></span>: 
( not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c22">nq3</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c21">nq2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c20">nq1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E10:4"/><span class="lab"><font color="Green" title="E10">A10</font></span>: 
( not <font color="Maroon" title="c10">ns1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c22">nq3</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c21">nq2</font></span>)</span>,<font color="Maroon" title="c20">nq1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E11:4"/><span class="lab"><font color="Green" title="E11">A11</font></span>: 
( not <font color="Maroon" title="c11">ns2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c22">nq3</font></span>)</span>,<font color="Maroon" title="c21">nq2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c20">nq1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E12:4"/><span class="lab"><font color="Green" title="E12">A12</font></span>: 
( not <font color="Maroon" title="c12">ns3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c22">nq3</font></span>)</span>,<font color="Maroon" title="c21">nq2</font>,<font color="Maroon" title="c20">nq1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E13:4"/><span class="lab"><font color="Green" title="E13">A13</font></span>: 
( not <font color="Maroon" title="c13">ns4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c22">nq3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c21">nq2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c20">nq1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E14:4"/><span class="lab"><font color="Green" title="E14">A14</font></span>: 
( not <font color="Maroon" title="c14">ns5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c22">nq3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c21">nq2</font></span>)</span>,<font color="Maroon" title="c20">nq1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E15:4"/><span class="lab"><font color="Green" title="E15">A15</font></span>: 
( not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c22">nq3</font>,<font color="Maroon" title="c21">nq2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c20">nq1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E16:4"/><span class="lab"><font color="Green" title="E16">A16</font></span>: 
( not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c22">nq3</font>,<font color="Maroon" title="c21">nq2</font>,<font color="Maroon" title="c20">nq1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E17:4"/><span class="lab"><font color="Green" title="E17">A17</font></span>: 
( not <font color="Maroon" title="c20">nq1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E18:4"/><span class="lab"><font color="Green" title="E18">A18</font></span>: 
( not <font color="Maroon" title="c21">nq2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c17">q1</font>,<font color="Maroon" title="c18">q2</font></span>)</span>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E19:4"/><span class="lab"><font color="Green" title="E19">A19</font></span>: 
( not <font color="Maroon" title="c22">nq3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c19">q3</font>,<span class="p3">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c17">q1</font>,<span class="p3">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">q2</font>,<font color="Maroon" title="c19">q3</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( not <font color="Maroon" title="c10">ns1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">s0</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">s0</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c10">ns1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c11">ns2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c11">ns2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c12">ns3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">s2</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">s2</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c12">ns3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c13">ns4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">s3</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">s3</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c13">ns4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c14">ns5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">s4</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">s4</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c14">ns5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">s5</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">s5</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">s6</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">s6</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c8">s7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c23">R</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c8">s7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c23">R</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )</span><br/>

<a NAME="E20:4"/>
( not <font color="Maroon" title="c10">ns1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff ( <font color="Maroon" title="c22">nq3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; (  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c17">q1</font>,<font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:4"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="txt" href="#E17:4"><span class="lab"><font color="Green" title="E17">A17</font></span></a>, <a class="txt" href="#E18:4"><span class="lab"><font color="Green" title="E18">A18</font></span></a></span>;<br/>
<a NAME="E21:4"/><span class="kw">then </span>
( not <font color="Maroon" title="c10">ns1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff ( ( <font color="Maroon" title="c19">q3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">q2</font>,<font color="Maroon" title="c19">q3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; <font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E19">A19</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:4"/>
( not <font color="Maroon" title="c10">ns1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">s0</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4"><span class="lab"><font color="Green" title="E1">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( not <font color="Maroon" title="c11">ns2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c11">ns2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c12">ns3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">s2</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">s2</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c12">ns3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c13">ns4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">s3</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">s3</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c13">ns4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c14">ns5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">s4</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">s4</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c14">ns5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">s5</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">s5</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">s6</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">s6</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c8">s7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c23">R</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c8">s7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c23">R</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )</span><br/>

<a NAME="E23:4"/>
( not <font color="Maroon" title="c11">ns2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff ( <font color="Maroon" title="c22">nq3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c17">q1</font>,<font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; (  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:4"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="txt" href="#E17:4"><span class="lab"><font color="Green" title="E17">A17</font></span></a>, <a class="txt" href="#E18:4"><span class="lab"><font color="Green" title="E18">A18</font></span></a></span>;<br/>
<a NAME="E24:4"/><span class="kw">then </span>
( not <font color="Maroon" title="c11">ns2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff ( <font color="Maroon" title="c22">nq3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c17">q1</font>,<font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )
 
;<br/>
<a NAME="E25:4"/><span class="kw">then </span>
( not <font color="Maroon" title="c11">ns2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff (  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c19">q3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c17">q1</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">q2</font>,<font color="Maroon" title="c19">q3</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E19">A19</font></span></a></span>;<br/>
<a NAME="E26:4"/><span class="kw">then </span>
( not <font color="Maroon" title="c11">ns2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff ( ( <font color="Maroon" title="c19">q3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; <font color="Maroon" title="c19">q3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E27:4"/>
( not <font color="Maroon" title="c11">ns2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4"><span class="lab"><font color="Green" title="E2">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( not <font color="Maroon" title="c12">ns3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">s2</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">s2</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c12">ns3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c13">ns4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">s3</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">s3</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c13">ns4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c14">ns5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">s4</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">s4</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c14">ns5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">s5</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">s5</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">s6</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">s6</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c8">s7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c23">R</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c8">s7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c23">R</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )</span><br/>

<a NAME="E28:4"/>
( not <font color="Maroon" title="c12">ns3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff ( <font color="Maroon" title="c22">nq3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c17">q1</font>,<font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:4"><span class="lab"><font color="Green" title="E12">A12</font></span></a>, <a class="txt" href="#E17:4"><span class="lab"><font color="Green" title="E17">A17</font></span></a>, <a class="txt" href="#E18:4"><span class="lab"><font color="Green" title="E18">A18</font></span></a></span>;<br/>
<a NAME="E29:4"/><span class="kw">then </span>
( not <font color="Maroon" title="c12">ns3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff ( ( <font color="Maroon" title="c19">q3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">q2</font>,<font color="Maroon" title="c19">q3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; not <font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E19">A19</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E30:4"/>
( not <font color="Maroon" title="c12">ns3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">s2</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:4"><span class="lab"><font color="Green" title="E3">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( not <font color="Maroon" title="c13">ns4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">s3</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">s3</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c13">ns4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c14">ns5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">s4</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">s4</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c14">ns5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">s5</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">s5</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">s6</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">s6</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c8">s7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c23">R</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c8">s7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c23">R</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )</span><br/>

<a NAME="E31:4"/>
( not <font color="Maroon" title="c13">ns4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff ( not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c19">q3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c17">q1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">q2</font>,<font color="Maroon" title="c19">q3</font></span>)</span></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c21">nq2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:4"><span class="lab"><font color="Green" title="E13">A13</font></span></a>, <a class="txt" href="#E17:4"><span class="lab"><font color="Green" title="E17">A17</font></span></a>, <a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E19">A19</font></span></a></span>;<br/>
<a NAME="E32:4"/><span class="kw">then </span>
( not <font color="Maroon" title="c13">ns4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff ( not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c19">q3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c17">q1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">q2</font>,<font color="Maroon" title="c19">q3</font></span>)</span></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c17">q1</font>,<font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:4"><span class="lab"><font color="Green" title="E18">A18</font></span></a></span>;<br/>
<a NAME="E33:4"/><span class="kw">then </span>
( not <font color="Maroon" title="c13">ns4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff ( ( ( not <font color="Maroon" title="c19">q3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) or ( not <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">q2</font>,<font color="Maroon" title="c19">q3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) ) &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )
 
;<br/>
<a NAME="E34:4"/><span class="kw">then </span>
( not <font color="Maroon" title="c13">ns4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff ( <font color="Maroon" title="c19">q3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E35:4"/>
( not <font color="Maroon" title="c13">ns4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">s3</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:4"><span class="lab"><font color="Green" title="E4">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( not <font color="Maroon" title="c14">ns5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">s4</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">s4</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c14">ns5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">s5</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">s5</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">s6</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">s6</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c8">s7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c23">R</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c8">s7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c23">R</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )</span><br/>

<a NAME="E36:4"/>
( not <font color="Maroon" title="c14">ns5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff ( not <font color="Maroon" title="c22">nq3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; (  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c17">q1</font>,<font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:4"><span class="lab"><font color="Green" title="E14">A14</font></span></a>, <a class="txt" href="#E17:4"><span class="lab"><font color="Green" title="E17">A17</font></span></a>, <a class="txt" href="#E18:4"><span class="lab"><font color="Green" title="E18">A18</font></span></a></span>;<br/>
<a NAME="E37:4"/><span class="kw">then </span>
( not <font color="Maroon" title="c14">ns5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff ( not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c19">q3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c17">q1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">q2</font>,<font color="Maroon" title="c19">q3</font></span>)</span></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E19">A19</font></span></a></span>;<br/>
<a NAME="E38:4"/><span class="kw">then </span>
( not <font color="Maroon" title="c14">ns5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff ( not <font color="Maroon" title="c19">q3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E39:4"/>
( not <font color="Maroon" title="c14">ns5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c5">s4</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:4"><span class="lab"><font color="Green" title="E5">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">s5</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">s5</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">s6</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">s6</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c8">s7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c23">R</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c8">s7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c23">R</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )</span><br/>

<a NAME="E40:4"/>
( not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff ( not <font color="Maroon" title="c22">nq3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c17">q1</font>,<font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; (  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:4"><span class="lab"><font color="Green" title="E15">A15</font></span></a>, <a class="txt" href="#E17:4"><span class="lab"><font color="Green" title="E17">A17</font></span></a>, <a class="txt" href="#E18:4"><span class="lab"><font color="Green" title="E18">A18</font></span></a></span>;<br/>
<a NAME="E41:4"/><span class="kw">then </span>
( not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff ( not <font color="Maroon" title="c22">nq3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c17">q1</font>,<font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )
 
;<br/>
<a NAME="E42:4"/><span class="kw">then </span>
( not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff ( ( ( not <font color="Maroon" title="c19">q3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) or ( not <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">q2</font>,<font color="Maroon" title="c19">q3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) ) &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E19">A19</font></span></a></span>;<br/>
<a NAME="E43:4"/><span class="kw">then </span>
( not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff ( not <font color="Maroon" title="c19">q3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E44:4"/>
( not <font color="Maroon" title="c15">ns6</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c6">s5</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:4"><span class="lab"><font color="Green" title="E6">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">s6</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">s6</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c8">s7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c23">R</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c8">s7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c23">R</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )</span><br/>

<a NAME="E45:4"/>
( not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff ( not <font color="Maroon" title="c22">nq3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c17">q1</font>,<font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:4"><span class="lab"><font color="Green" title="E16">A16</font></span></a>, <a class="txt" href="#E17:4"><span class="lab"><font color="Green" title="E17">A17</font></span></a>, <a class="txt" href="#E18:4"><span class="lab"><font color="Green" title="E18">A18</font></span></a></span>;<br/>
<a NAME="E46:4"/><span class="kw">then </span>
( not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff ( not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c19">q3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c17">q1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">q2</font>,<font color="Maroon" title="c19">q3</font></span>)</span></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E19">A19</font></span></a></span>;<br/>
<a NAME="E47:4"/><span class="kw">then </span>
( not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff ( not <font color="Maroon" title="c19">q3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E48:4"/>
( not <font color="Maroon" title="c16">ns7</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c7">s6</font>,<font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:4"><span class="lab"><font color="Green" title="E7">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c8">s7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c23">R</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span><br/>

<a NAME="E49:4"/>
( ( ( not <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c17">q1</font>,<font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c19">q3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c17">q1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">q2</font>,<font color="Maroon" title="c19">q3</font></span>)</span></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) or <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) iff  not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:4"><span class="lab"><font color="Green" title="E9">A9</font></span></a>, <a class="txt" href="#E17:4"><span class="lab"><font color="Green" title="E17">A17</font></span></a>, <a class="txt" href="#E18:4"><span class="lab"><font color="Green" title="E18">A18</font></span></a>, <a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E19">A19</font></span></a></span>;<br/>
<a NAME="E50:4"/><span class="kw">then </span>
( ( ( not <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c19">q3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c17">q1</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c18">q2</font>,<font color="Maroon" title="c19">q3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) or <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) iff  not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 
;<br/>
<a NAME="E51:4"/><span class="kw">then </span>
( ( ( not <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c18">q2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c17">q1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c19">q3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) or <font color="Maroon" title="c23">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) iff  not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 
;<br/>
<span class="kw">hence </span><a NAME="E52:4"/>
( not <font color="Maroon" title="c9">ns0</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c8">s7</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c23">R</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:4"><span class="lab"><font color="Green" title="E8">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
