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;
 reserve P for non empty ProofSystem;
 reserve B, B1, B2 for Subset of P;
 reserve F for finite Subset of P;

theorem Th59:
  for P, B, B1, B2 st B1 is B-omitting & B2 c= B1 holds B2 is B-omitting
proof
  let P, B, B1, B2;
  set A = the Axioms of P;
  set R = the Rules of P;
  assume that A1: B1 is B-omitting and A2: B2 c= B1;
  consider a such that A3: a in B and A4: not P \/ B1 |- a by A1;
  take a;
  A \/ B1 is Extension of A \/ B2 & R is Extension of R
    by A2, Def11, Def12, XBOOLE_1:9;
  hence thesis by A3, A4, Th54;
end;
