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