theorem :: GATE_2:4

for s0, s1, s2, s3, s4, s5, s6, s7, ns0, ns1, ns2, ns3, ns4, ns5, ns6, ns7, q1, q2, q3, nq1, nq2, nq3, R being set holds
not ( ( not s0 is empty implies not AND3 ((NOT1 q3),(NOT1 q2),(NOT1 q1)) is empty ) & ( not AND3 ((NOT1 q3),(NOT1 q2),(NOT1 q1)) is empty implies not s0 is empty ) & ( not s1 is empty implies not AND3 ((NOT1 q3),(NOT1 q2),q1) is empty ) & ( not AND3 ((NOT1 q3),(NOT1 q2),q1) is empty implies not s1 is empty ) & ( not s2 is empty implies not AND3 ((NOT1 q3),q2,(NOT1 q1)) is empty ) & ( not AND3 ((NOT1 q3),q2,(NOT1 q1)) is empty implies not s2 is empty ) & ( not s3 is empty implies not AND3 ((NOT1 q3),q2,q1) is empty ) & ( not AND3 ((NOT1 q3),q2,q1) is empty implies not s3 is empty ) & ( not s4 is empty implies not AND3 (q3,(NOT1 q2),(NOT1 q1)) is empty ) & ( not AND3 (q3,(NOT1 q2),(NOT1 q1)) is empty implies not s4 is empty ) & ( not s5 is empty implies not AND3 (q3,(NOT1 q2),q1) is empty ) & ( not AND3 (q3,(NOT1 q2),q1) is empty implies not s5 is empty ) & ( not s6 is empty implies not AND3 (q3,q2,(NOT1 q1)) is empty ) & ( not AND3 (q3,q2,(NOT1 q1)) is empty implies not s6 is empty ) & ( not s7 is empty implies not AND3 (q3,q2,q1) is empty ) & ( not AND3 (q3,q2,q1) is empty implies not s7 is empty ) & ( not ns0 is empty implies not AND3 ((NOT1 nq3),(NOT1 nq2),(NOT1 nq1)) is empty ) & ( not AND3 ((NOT1 nq3),(NOT1 nq2),(NOT1 nq1)) is empty implies not ns0 is empty ) & ( not ns1 is empty implies not AND3 ((NOT1 nq3),(NOT1 nq2),nq1) is empty ) & ( not AND3 ((NOT1 nq3),(NOT1 nq2),nq1) is empty implies not ns1 is empty ) & ( not ns2 is empty implies not AND3 ((NOT1 nq3),nq2,(NOT1 nq1)) is empty ) & ( not AND3 ((NOT1 nq3),nq2,(NOT1 nq1)) is empty implies not ns2 is empty ) & ( not ns3 is empty implies not AND3 ((NOT1 nq3),nq2,nq1) is empty ) & ( not AND3 ((NOT1 nq3),nq2,nq1) is empty implies not ns3 is empty ) & ( not ns4 is empty implies not AND3 (nq3,(NOT1 nq2),(NOT1 nq1)) is empty ) & ( not AND3 (nq3,(NOT1 nq2),(NOT1 nq1)) is empty implies not ns4 is empty ) & ( not ns5 is empty implies not AND3 (nq3,(NOT1 nq2),nq1) is empty ) & ( not AND3 (nq3,(NOT1 nq2),nq1) is empty implies not ns5 is empty ) & ( not ns6 is empty implies not AND3 (nq3,nq2,(NOT1 nq1)) is empty ) & ( not AND3 (nq3,nq2,(NOT1 nq1)) is empty implies not ns6 is empty ) & ( not ns7 is empty implies not AND3 (nq3,nq2,nq1) is empty ) & ( not AND3 (nq3,nq2,nq1) is empty implies not ns7 is empty ) & ( not nq1 is empty implies not AND2 ((NOT1 q1),R) is empty ) & ( not AND2 ((NOT1 q1),R) is empty implies not nq1 is empty ) & ( not nq2 is empty implies not AND2 ((XOR2 (q1,q2)),R) is empty ) & ( not AND2 ((XOR2 (q1,q2)),R) is empty implies not nq2 is empty ) & ( not nq3 is empty implies not AND2 ((OR2 ((AND2 (q3,(NOT1 q1))),(AND2 (q1,(XOR2 (q2,q3)))))),R) is empty ) & ( not AND2 ((OR2 ((AND2 (q3,(NOT1 q1))),(AND2 (q1,(XOR2 (q2,q3)))))),R) is empty implies not nq3 is empty ) & not ( ( not ns1 is empty implies not AND2 (s0,R) is empty ) & ( not AND2 (s0,R) is empty implies not ns1 is empty ) & ( not ns2 is empty implies not AND2 (s1,R) is empty ) & ( not AND2 (s1,R) is empty implies not ns2 is empty ) & ( not ns3 is empty implies not AND2 (s2,R) is empty ) & ( not AND2 (s2,R) is empty implies not ns3 is empty ) & ( not ns4 is empty implies not AND2 (s3,R) is empty ) & ( not AND2 (s3,R) is empty implies not ns4 is empty ) & ( not ns5 is empty implies not AND2 (s4,R) is empty ) & ( not AND2 (s4,R) is empty implies not ns5 is empty ) & ( not ns6 is empty implies not AND2 (s5,R) is empty ) & ( not AND2 (s5,R) is empty implies not ns6 is empty ) & ( not ns7 is empty implies not AND2 (s6,R) is empty ) & ( not AND2 (s6,R) is empty implies not ns7 is empty ) & ( not ns0 is empty implies not OR2 (s7,(NOT1 R)) is empty ) & ( not OR2 (s7,(NOT1 R)) is empty implies not ns0 is empty ) ) )