reserve k,m,n for Element of NAT,
  i, j for Nat,
  a, b, c for object,
  X, Y, Z for set,
  D, D1, D2 for non empty set;
reserve p, q, r, s for FinSequence;
reserve t, u, v, w for GRZ-formula;
reserve R, R1, R2 for GRZ-rule;
reserve A, A1, A2 for non empty Subset of GRZ-formula-set;
reserve B, B1, B2 for Subset of GRZ-formula-set;
reserve P, P1, P2 for GRZ-formula-sequence;
reserve S, S1, S2 for GRZ-formula-finset;

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;
