reserve k,m,n for Element of NAT,
  i, j for Nat,
  a, b, c for object,
  X, Y, Z for set,
  D, D1, D2 for non empty set;
reserve p, q, r, s for FinSequence;
reserve t, u, v, w for GRZ-formula;
reserve R, R1, R2 for GRZ-rule;
reserve A, A1, A2 for non empty Subset of GRZ-formula-set;
reserve B, B1, B2 for Subset of GRZ-formula-set;
reserve P, P1, P2 for GRZ-formula-sequence;
reserve S, S1, S2 for GRZ-formula-finset;

theorem Th47:
  for A, R, S st A, R |- S ex S1 st S c= S1 & S1 is (A, R)-correct
proof
  let A, R, S;
  assume A1: A, R |- S;
  defpred X[ set ] means ex S1 st $1 c= S1 & S1 is (A, R)-correct;
  A2: S is finite;
  A10: X[ {} ]
  proof
    reconsider t = the Element of A as GRZ-formula;
    consider S1 such that t in S1 and A11: S1 is (A, R)-correct by Th45, Th46;
    take S1;
    thus thesis by A11;
  end;
  A20: for x, B being set st x in S & B c= S & X[ B ] holds X[ B \/ {x} ]
  proof
    let x, B be set;
    assume that
      A21: x in S and
           B c= S and
      A23: X[ B ];
    reconsider t = x as GRZ-formula by A21;
    consider S1 such that
      A24: t in S1 and
      A25: S1 is (A, R)-correct by A1, A21, Th45;
    consider S2 such that
      A26: B c= S2 and
      A27: S2 is (A, R)-correct by A23;
    take S1 \/ S2;
    {x} c= S1 by A24, TARSKI:def 1;
    hence B \/ {x} c= S1 \/ S2 by A26, XBOOLE_1:13;
    thus S1 \/ S2 is (A, R)-correct by A25, A27, Th43;
  end;
  thus X[ S ] from FINSET_1:sch 2(A2, A10, A20);
end;
