 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
  PrimeIdeals(A,J1) = PrimeIdeals(A,J2) iff sqrt J1 = sqrt J2
   proof
     thus PrimeIdeals(A,J1) = PrimeIdeals(A,J2) implies sqrt J1 = sqrt J2
     proof
       assume
A2:    PrimeIdeals(A,J1) = PrimeIdeals(A,J2);
       sqrt J1 = meet PrimeIdeals(A,J1) by Th28
       .= sqrt J2 by Th28,A2;
       hence thesis;
     end;
     assume
A3:  sqrt J1 = sqrt J2;
     PrimeIdeals(A,J1) = PrimeIdeals(A,J1-Ideal) by IDEAL_1:44
       .= PrimeIdeals(A,sqrt(J1-Ideal)) by Th37
       .= PrimeIdeals(A,sqrt J1) by IDEAL_1:44
       .= PrimeIdeals(A,sqrt(J2-Ideal)) by A3,IDEAL_1:44
       .= PrimeIdeals(A,J2-Ideal) by Th37
       .= PrimeIdeals(A,J2) by IDEAL_1:44;
     hence thesis;
   end;
