 reserve R for commutative Ring;
 reserve A,B for non degenerated commutative Ring;
 reserve h for Function of A,B;
 reserve I0,I,I1,I2 for Ideal of A;
 reserve J,J1,J2 for proper Ideal of A;
 reserve p for prime Ideal of A;
 reserve S,S1 for non empty Subset of A;
 reserve E,E1,E2 for Subset of A;
 reserve a,b,f for Element of A;
 reserve n for Nat;
 reserve x,o,o1 for object;

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;
