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;
