 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 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;
