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 Th43:
  for A, R, S1, S2 st S1 is (A, R)-correct & S2 is (A, R)-correct holds
      S1 \/ S2 is (A, R)-correct
proof
  let A, R, S1, S2;
  assume
    A1: S1 is (A, R)-correct & S2 is (A, R)-correct;
  consider P1, P2 such that
    A3: P1 is (A, R)-correct and
    A4: S1 = rng P1 and
    A5: P2 is (A, R)-correct and
    A6: S2 = rng P2 by A1;
    reconsider S = rng(P1^P2) as GRZ-formula-finset;
    S = S1 \/ S2 by A4, A6, FINSEQ_1:31;
    hence thesis by A3, A5, Th42;
end;
