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 Th56:
  for P, B, B1 st B is inconsistent holds B1 is consistent iff B1 is B-omitting
proof
  let P, B, B1;
  set A = the Axioms of P;
  set R = the Rules of P;
  assume B is inconsistent;
  then A1: P \/ B is inconsistent;
  thus B1 is consistent implies B1 is B-omitting
  proof
    assume B1 is consistent;
    then P \/ B1 is consistent;
    then consider a such that A2: a in P \/ B1 and A3: not P \/ B1 |- a;
    a in P \/ B by A2;
    then A5: P \/ B |- a by A1;
    assume A7: B1 is non B-omitting;
    for b st b in A \/ B holds A \/ B1, R |- b
      proof
      let b;
      assume b in A \/ B;
      then per cases by XBOOLE_0:def 3;
        suppose b in A;
        then b in A \/ B1 by XBOOLE_0:def 3;
        hence thesis by Th46;
        end;
        suppose b in B;
        then P \/ B1 |- b by A7;
        hence thesis;
        end;
      end;
    hence thesis by A3, A5, Th53;
  end;
  assume B1 is B-omitting;
  then consider a such that A11: a in B and A12: not P \/ B1 |- a;
  a in P \/ B1 by A11;
  then P \/ B1 is consistent by A12;
  hence thesis;
end;
