 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 Th37:
  PrimeIdeals(A,sqrt(S-Ideal)) = PrimeIdeals(A,S-Ideal)
   proof
     thus PrimeIdeals(A,sqrt(S-Ideal)) c= PrimeIdeals(A,S-Ideal)
     proof
       let p be object;
       assume p in PrimeIdeals(A,sqrt(S-Ideal)); then
       consider p1 be prime Ideal of A such that
A3:    p1 = p and
A4:    sqrt(S-Ideal) c= p1;
       S-Ideal c= p1 by A4,Th36;
       hence p in PrimeIdeals(A,S-Ideal) by A3;
     end;
     let p be object;
     assume p in PrimeIdeals(A,S-Ideal); then
     consider p1 be prime Ideal of A such that
A7:  p1 = p and
A8:  S-Ideal c= p1;
     sqrt(S-Ideal) c= p1 by A8,Th35;
     hence p in PrimeIdeals(A,sqrt(S-Ideal)) by A7;
   end;
