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
  for P, a holds P |- a iff for X being P-closed set holds a in X
proof
  let P, a;
  set A = the Axioms of P;
  thus P |- a implies for X being P-closed set holds a in X by Th65;
  assume for X being P-closed set holds a in X;
  then a in Theorems P;
  hence thesis by Def30r;
end;
