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;
