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;
