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
  for A, R, P, P1, P2 st P is (A, R)-correct & P = P1^P2 holds
      P1 is (A, R)-correct
proof
  let A, R, P, P1, P2;
  assume that
    A1: P is (A, R)-correct and
    A2: P = P1^P2;
  set P0 = <*>GRZ-formula-set;
  let k;
  assume A3: k in dom P1;
  dom P1 c= dom P by A2, FINSEQ_1:26;
  then P1^P0, k is_a_correct_step_wrt A, R by A1, A2, A3, Lm41;
  hence P1, k is_a_correct_step_wrt A, R by FINSEQ_1:34;
end;
