reserve i, j, k, l, m, n for Nat,
  a, b, c, t, u for object,
  X, Y, Z for set,
  D, D1, D2, Fml for non empty set;
reserve p, q, r, s for FinSequence;
 reserve R, R1, R2 for Rule;
 reserve A, A1, A2 for non empty set;
 reserve B, B1, B2 for set;
 reserve P, P1, P2 for Formula-sequence;
 reserve S, S1, S2 for Formula-finset;

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