theorem Th38:
  E2 c= E1 implies PrimeIdeals(A,E1) c= PrimeIdeals(A,E2)
  proof
    assume
A1: E2 c= E1;
    let x;
    assume x in PrimeIdeals(A,E1); then
    consider x1 be prime Ideal of A such that
A3: x1 = x and
A4: E1 c= x1;
    E2 c= x1 by A1,A4;
    hence thesis by A3;
  end;
