 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;
 reserve m for maximal Ideal of A;
 reserve p for prime Ideal of A;
 reserve k for non zero Nat;

theorem Th32:
  PrimeIdeals(A,S) c= Ideals(A,S)
  proof
    let x be object;
    assume x in PrimeIdeals(A,S); then
    consider x1 be prime Ideal of A such that
A2: x1 = x and
A3: S c= x1;
    thus thesis by A2, A3;
  end;
