theorem Th2:
  Ideals(A,S) = Ideals(A,S-Ideal)
  proof
    thus Ideals(A,S) c= Ideals(A,S-Ideal)
    proof
      let x be object;
      assume x in Ideals(A,S); then
      consider y being Ideal of A such that
A2:   x = y and
A3:   S c= y;
      S-Ideal c= y-Ideal by A3,IDEAL_1:57; then
      S-Ideal c= y by IDEAL_1:44;
      hence thesis by A2;
    end;
      let x be object;
      assume x in Ideals(A,S-Ideal); then
      consider y being Ideal of A such that
A8:   x = y and
A9:   S-Ideal c= y;
      S c= S-Ideal by IDEAL_1:def 14; then
      S c= y by A9;
      hence thesis by A8;
  end;
