Article GATE_2, MML version 4.99.1005
:: GATE_2:th 1
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22 being set
st (b1 is not empty implies AND3(NOT1 b19,NOT1 b18,NOT1 b17) is not empty) &
(AND3(NOT1 b19,NOT1 b18,NOT1 b17) is not empty implies b1 is not empty) &
(b2 is not empty implies AND3(NOT1 b19,NOT1 b18,b17) is not empty) &
(AND3(NOT1 b19,NOT1 b18,b17) is not empty implies b2 is not empty) &
(b3 is not empty implies AND3(NOT1 b19,b18,NOT1 b17) is not empty) &
(AND3(NOT1 b19,b18,NOT1 b17) is not empty implies b3 is not empty) &
(b4 is not empty implies AND3(NOT1 b19,b18,b17) is not empty) &
(AND3(NOT1 b19,b18,b17) is not empty implies b4 is not empty) &
(b5 is not empty implies AND3(b19,NOT1 b18,NOT1 b17) is not empty) &
(AND3(b19,NOT1 b18,NOT1 b17) is not empty implies b5 is not empty) &
(b6 is not empty implies AND3(b19,NOT1 b18,b17) is not empty) &
(AND3(b19,NOT1 b18,b17) is not empty implies b6 is not empty) &
(b7 is not empty implies AND3(b19,b18,NOT1 b17) is not empty) &
(AND3(b19,b18,NOT1 b17) is not empty implies b7 is not empty) &
(b8 is not empty implies AND3(b19,b18,b17) is not empty) &
(AND3(b19,b18,b17) is not empty implies b8 is not empty) &
(b9 is not empty implies AND3(NOT1 b22,NOT1 b21,NOT1 b20) is not empty) &
(AND3(NOT1 b22,NOT1 b21,NOT1 b20) is not empty implies b9 is not empty) &
(b10 is not empty implies AND3(NOT1 b22,NOT1 b21,b20) is not empty) &
(AND3(NOT1 b22,NOT1 b21,b20) is not empty implies b10 is not empty) &
(b11 is not empty implies AND3(NOT1 b22,b21,NOT1 b20) is not empty) &
(AND3(NOT1 b22,b21,NOT1 b20) is not empty implies b11 is not empty) &
(b12 is not empty implies AND3(NOT1 b22,b21,b20) is not empty) &
(AND3(NOT1 b22,b21,b20) is not empty implies b12 is not empty) &
(b13 is not empty implies AND3(b22,NOT1 b21,NOT1 b20) is not empty) &
(AND3(b22,NOT1 b21,NOT1 b20) is not empty implies b13 is not empty) &
(b14 is not empty implies AND3(b22,NOT1 b21,b20) is not empty) &
(AND3(b22,NOT1 b21,b20) is not empty implies b14 is not empty) &
(b15 is not empty implies AND3(b22,b21,NOT1 b20) is not empty) &
(AND3(b22,b21,NOT1 b20) is not empty implies b15 is not empty) &
(b16 is not empty implies AND3(b22,b21,b20) is not empty) &
(AND3(b22,b21,b20) is not empty implies b16 is not empty) &
(b20 is not empty implies NOT1 b17 is not empty) &
(NOT1 b17 is not empty implies b20 is not empty) &
(b21 is not empty implies XOR2(b17,b18) is not empty) &
(XOR2(b17,b18) is not empty implies b21 is not empty) &
(b22 is not empty implies OR2(AND2(b19,NOT1 b17),AND2(b17,XOR2(b18,b19))) is not empty) &
(OR2(AND2(b19,NOT1 b17),AND2(b17,XOR2(b18,b19))) is not empty implies b22 is not empty)
holds (b10 is not empty implies b1 is not empty) & (b1 is not empty implies b10 is not empty) & (b11 is not empty implies b2 is not empty) & (b2 is not empty implies b11 is not empty) & (b12 is not empty implies b3 is not empty) & (b3 is not empty implies b12 is not empty) & (b13 is not empty implies b4 is not empty) & (b4 is not empty implies b13 is not empty) & (b14 is not empty implies b5 is not empty) & (b5 is not empty implies b14 is not empty) & (b15 is not empty implies b6 is not empty) & (b6 is not empty implies b15 is not empty) & (b16 is not empty implies b7 is not empty) & (b7 is not empty implies b16 is not empty) & (b9 is not empty implies b8 is not empty) & (b8 is not empty implies b9 is not empty);
:: GATE_2:th 2
theorem
for b1, b2, b3, b4 being set holds
AND3(AND2(b1,b2),AND2(b3,b2),AND2(b4,b2)) is not empty
iff
AND2(AND3(b1,b3,b4),b2) is not empty;
:: GATE_2:th 3
theorem
for b1, b2, b3, b4 being set holds
(AND2(b1,b2) is empty implies OR2(NOT1 b1,NOT1 b2) is not empty) &
(OR2(NOT1 b1,NOT1 b2) is empty or AND2(b1,b2) is empty) &
(OR2(b1,b2) is not empty & OR2(b3,b2) is not empty implies OR2(AND2(b1,b3),b2) is not empty) &
(OR2(AND2(b1,b3),b2) is empty or OR2(b1,b2) is not empty & OR2(b3,b2) is not empty) &
(OR2(b1,b2) is not empty & OR2(b3,b2) is not empty & OR2(b4,b2) is not empty implies OR2(AND3(b1,b3,b4),b2) is not empty) &
(OR2(AND3(b1,b3,b4),b2) is empty or OR2(b1,b2) is not empty & OR2(b3,b2) is not empty & OR2(b4,b2) is not empty) &
(OR2(b1,b2) is not empty & (b1 is not empty implies b3 is not empty) & (b3 is not empty implies b1 is not empty) implies OR2(b3,b2) is not empty);
:: GATE_2:th 4
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23 being set
st (b1 is not empty implies AND3(NOT1 b19,NOT1 b18,NOT1 b17) is not empty) &
(AND3(NOT1 b19,NOT1 b18,NOT1 b17) is not empty implies b1 is not empty) &
(b2 is not empty implies AND3(NOT1 b19,NOT1 b18,b17) is not empty) &
(AND3(NOT1 b19,NOT1 b18,b17) is not empty implies b2 is not empty) &
(b3 is not empty implies AND3(NOT1 b19,b18,NOT1 b17) is not empty) &
(AND3(NOT1 b19,b18,NOT1 b17) is not empty implies b3 is not empty) &
(b4 is not empty implies AND3(NOT1 b19,b18,b17) is not empty) &
(AND3(NOT1 b19,b18,b17) is not empty implies b4 is not empty) &
(b5 is not empty implies AND3(b19,NOT1 b18,NOT1 b17) is not empty) &
(AND3(b19,NOT1 b18,NOT1 b17) is not empty implies b5 is not empty) &
(b6 is not empty implies AND3(b19,NOT1 b18,b17) is not empty) &
(AND3(b19,NOT1 b18,b17) is not empty implies b6 is not empty) &
(b7 is not empty implies AND3(b19,b18,NOT1 b17) is not empty) &
(AND3(b19,b18,NOT1 b17) is not empty implies b7 is not empty) &
(b8 is not empty implies AND3(b19,b18,b17) is not empty) &
(AND3(b19,b18,b17) is not empty implies b8 is not empty) &
(b9 is not empty implies AND3(NOT1 b22,NOT1 b21,NOT1 b20) is not empty) &
(AND3(NOT1 b22,NOT1 b21,NOT1 b20) is not empty implies b9 is not empty) &
(b10 is not empty implies AND3(NOT1 b22,NOT1 b21,b20) is not empty) &
(AND3(NOT1 b22,NOT1 b21,b20) is not empty implies b10 is not empty) &
(b11 is not empty implies AND3(NOT1 b22,b21,NOT1 b20) is not empty) &
(AND3(NOT1 b22,b21,NOT1 b20) is not empty implies b11 is not empty) &
(b12 is not empty implies AND3(NOT1 b22,b21,b20) is not empty) &
(AND3(NOT1 b22,b21,b20) is not empty implies b12 is not empty) &
(b13 is not empty implies AND3(b22,NOT1 b21,NOT1 b20) is not empty) &
(AND3(b22,NOT1 b21,NOT1 b20) is not empty implies b13 is not empty) &
(b14 is not empty implies AND3(b22,NOT1 b21,b20) is not empty) &
(AND3(b22,NOT1 b21,b20) is not empty implies b14 is not empty) &
(b15 is not empty implies AND3(b22,b21,NOT1 b20) is not empty) &
(AND3(b22,b21,NOT1 b20) is not empty implies b15 is not empty) &
(b16 is not empty implies AND3(b22,b21,b20) is not empty) &
(AND3(b22,b21,b20) is not empty implies b16 is not empty) &
(b20 is not empty implies AND2(NOT1 b17,b23) is not empty) &
(AND2(NOT1 b17,b23) is not empty implies b20 is not empty) &
(b21 is not empty implies AND2(XOR2(b17,b18),b23) is not empty) &
(AND2(XOR2(b17,b18),b23) is not empty implies b21 is not empty) &
(b22 is not empty implies AND2(OR2(AND2(b19,NOT1 b17),AND2(b17,XOR2(b18,b19))),b23) is not empty) &
(AND2(OR2(AND2(b19,NOT1 b17),AND2(b17,XOR2(b18,b19))),b23) is not empty implies b22 is not empty)
holds (b10 is not empty implies AND2(b1,b23) is not empty) & (AND2(b1,b23) is not empty implies b10 is not empty) & (b11 is not empty implies AND2(b2,b23) is not empty) & (AND2(b2,b23) is not empty implies b11 is not empty) & (b12 is not empty implies AND2(b3,b23) is not empty) & (AND2(b3,b23) is not empty implies b12 is not empty) & (b13 is not empty implies AND2(b4,b23) is not empty) & (AND2(b4,b23) is not empty implies b13 is not empty) & (b14 is not empty implies AND2(b5,b23) is not empty) & (AND2(b5,b23) is not empty implies b14 is not empty) & (b15 is not empty implies AND2(b6,b23) is not empty) & (AND2(b6,b23) is not empty implies b15 is not empty) & (b16 is not empty implies AND2(b7,b23) is not empty) & (AND2(b7,b23) is not empty implies b16 is not empty) & (b9 is not empty implies OR2(b8,NOT1 b23) is not empty) & (OR2(b8,NOT1 b23) is not empty implies b9 is not empty);