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 Th65:
  for X, R, a holds
    X, R |- a iff for Y being R-closed X-extending set holds a in Y
proof
  let X, R, a;
  thus X,R |- a implies for Y being R-closed X-extending set holds a in Y
  proof
    assume A1: X, R |- a;
    let Y be R-closed X-extending set;
    defpred S[object] means $1 in Y;
    Y is X-extending;
    then A2: for b st b in X holds S[b];
    A3: for Z, b st [Z,b] in R & for c st c in Z holds S[c] holds S[b]
    proof
      let Z, b;
      assume that
        A5: [Z,b] in R and
        A6: for c st c in Z holds S[c];
      Z c= Y by A6;
      hence thesis by A5, Def28;
    end;
    for b st X, R |- b holds S[b] from ProofInduction(A2, A3);
    hence thesis by A1;
  end;
  assume for Y being R-closed X-extending set holds a in Y;
  then a in Theorems(X,R);
  hence thesis by Def30r;
end;
