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 Th61:
  for P, B, F st B is F-omitting ex B1 st B c= B1 & B1 is F-maximally-omitting
proof
  let P, B, F;
  assume A1: B is F-omitting;
  set A = the Axioms of P;
  set R = the Rules of P;
  set Q = P \/ B;
  reconsider G = F as Subset of Q;
  set Om = Omit(Q,G);
  A2: for C1 being Subset of Q, B1 st B1 = C1 \/ B holds
      (C1 in Om iff B1 is F-omitting)
  proof
    let C1 be Subset of Q;
    let B1;
    assume B1 = C1 \/ B;
    then A3: the Axioms of (Q \/ C1) = A \/ B1 by XBOOLE_1:4;
    thus C1 in Om implies B1 is F-omitting
    proof
      assume C1 in Om;
      then C1 is G-omitting by Def17;
      then consider a such that A4: a in G & not Q \/ C1 |- a;
      thus thesis by A3, A4;
    end;
    assume B1 is F-omitting;
    then consider a such that A6: a in F & not P \/ B1 |- a;
    C1 is G-omitting by A3, A6;
    hence thesis;
  end;
  A10: Om is non empty
  proof
    set E = {}(the carrier of Q);
    B = E \/ B;
    hence thesis by A1, A2;
  end;
  then consider Y being Element of Om such that
    A11:  for Z being Element of Om holds not Y c< Z by Th60;
  A12: Y in Om by A10;
  then reconsider B1 = B \/ Y as Subset of P by XBOOLE_1:8;
  take B1;
  thus B c= B1 by XBOOLE_1:7;
  thus B1 is F-omitting by A2, A12;
  let B2;
  reconsider C2 = B2 as Subset of Q;
  assume A15: B1 c< B2;
  assume A16: B2 is F-omitting;
  B c< B2 by A15, XBOOLE_1:7, XBOOLE_1:59;
  then B2 = B \/ C2 by XBOOLE_1:12;
  then A18: C2 in Om by A2, A16;
  Y c= B1 by XBOOLE_1:7;
  hence thesis by A11, A15, A18, XBOOLE_1:59;
end;
