theorem Th33:
  PrimeIdeals(A,S) = Ideals(A,S) /\ Spectrum A
   proof
     thus PrimeIdeals(A,S) c= Ideals(A,S) /\ Spectrum A
     proof
       let x be object;
       assume
A2:    x in PrimeIdeals(A,S); then
       consider x1 be prime Ideal of A such that
A3:    x1 = x and
       S c= x1;
A4:    x in Spectrum A by A3;
       x in Ideals(A,S) by A2,Th32,TARSKI:def 3;
       hence thesis by A4,XBOOLE_0:def 4;
     end;
     let x be object;
     assume x in Ideals(A,S) /\ Spectrum A; then
A8:  x in Ideals(A,S) & x in Spectrum A by XBOOLE_0:def 4; then
     consider x1 be Ideal of A such that
A10: x1 = x and
A11: S c= x1;
     x1 is prime Ideal of A by A8,Th22,A10;
     hence thesis by A10,A11;
   end;
