theorem :: GATE_2:3

for a, b, c, d being set holds
( ( AND2 (a,b) is empty implies not OR2 ((NOT1 a),(NOT1 b)) is empty ) & ( not OR2 ((NOT1 a),(NOT1 b)) is empty implies AND2 (a,b) is empty ) & ( not OR2 (a,b) is empty & not OR2 (c,b) is empty implies not OR2 ((AND2 (a,c)),b) is empty ) & ( not OR2 ((AND2 (a,c)),b) is empty implies ( not OR2 (a,b) is empty & not OR2 (c,b) is empty ) ) & ( not OR2 (a,b) is empty & not OR2 (c,b) is empty & not OR2 (d,b) is empty implies not OR2 ((AND3 (a,c,d)),b) is empty ) & ( not OR2 ((AND3 (a,c,d)),b) is empty implies ( not OR2 (a,b) is empty & not OR2 (c,b) is empty & not OR2 (d,b) is empty ) ) & not ( not OR2 (a,b) is empty & ( not a is empty implies not c is empty ) & ( not c is empty implies not a is empty ) & OR2 (c,b) is empty ) )