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;
