
theorem Th19:
  for n being Nat, x,y being FinSequence, p being set holds
  n-BitBorrowOutput(x,y) <> [p, and2] & n-BitBorrowOutput(x,y) <> [p, and2a] &
  n-BitBorrowOutput(x,y) <> [p, 'xor']
proof
  let n be Nat, x,y be FinSequence, p be set;
A1: dom and2 = 2-tuples_on BOOLEAN by FUNCT_2:def 1;
A2: dom and2a = 2-tuples_on BOOLEAN by FUNCT_2:def 1;
A3: dom 'xor' = 2-tuples_on BOOLEAN by FUNCT_2:def 1;
A4: proj1 [p, and2]`2 = 2-tuples_on BOOLEAN by A1;
A5: proj1 [p, and2a]`2 = 2-tuples_on BOOLEAN by A2;
A6: proj1 [p, 'xor']`2 = 2-tuples_on BOOLEAN by A3;
  proj1 (n-BitBorrowOutput(x,y))`2 = 0-tuples_on BOOLEAN or
  proj1 (n-BitBorrowOutput(x,y))`2 = 3-tuples_on BOOLEAN by Th18;
  hence thesis by A4,A5,A6,FINSEQ_2:110;
end;
