theorem Th45:
  for A, R, t holds A, R |- t iff ex S st t in S & S is (A, R)-correct
proof
  let A, R, t;
  thus A, R |- t implies ex S st t in S & S is (A, R)-correct
    proof
    assume A, R |- t;
    then consider P such that A1: t in rng P and A2: P is (A, R)-correct;
    take S = rng P;
    thus thesis by A1, A2;
    end;
  given S such that A10: t in S and A11: S is (A, R)-correct;
  consider P such that A12: S = rng P and A13: P is (A, R)-correct by A11;
  thus thesis by A10, A12, A13;
end;
