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;
 reserve C for Extension of B;
 reserve E for Extension of R;

theorem Th54:
  for B, C, R, E, a st B, R |- a holds C, E |- a
proof
  let B,C,R,E,a;
  C is B-extending & E is R-extending;
  hence thesis by Th44;
end;
