 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;
 reserve M0 for Ideal of B;
 reserve M0 for prime Ideal of B;

theorem Th52:
  h is RingHomomorphism implies
    (Spec h)"PrimeIdeals(A,E) = PrimeIdeals(B,h.:E)
    proof
      assume
A1:   h is RingHomomorphism;
      thus (Spec h)"PrimeIdeals(A,E) c= PrimeIdeals(B,h.:E)
      proof
        let x;
        assume
A3:     x in (Spec h)"PrimeIdeals(A,E); then
A4:     x in dom (Spec h) & (Spec h).x in PrimeIdeals(A,E) by FUNCT_1:def 7;
        reconsider x1 = x as Point of ZariskiTS B by A3;
        h"x1 in {p where p is prime Ideal of A: E c=p } by A1, Def9, A4; then
        consider q be prime Ideal of A such that
A6:     q = h"x1 and
A7:     E c= q;
A8:     h.:E c= h.:(h"x1) by A6,A7,RELAT_1:123;
        h.:(h"x1) c= x1 by FUNCT_1:75; then
A9:     h.:E c= x1 by A8;
        x1 in the carrier of ZariskiTS B; then
        x1 in Spectrum B by Def7; then
        x1 is prime Ideal of B by Th22;
        hence thesis by A9;
      end;
      let x;
        assume x in PrimeIdeals(B,h.:E); then
        consider q be prime Ideal of B such that
A12:    x = q and
A13:    h.:E c= q;
A14:    h"(h.:E) c= h"q by A13,RELAT_1:143;
        E c= h"(h.:E) by FUNCT_2:42; then
A15:    E c= h"q by A14;
        h"q is prime Ideal of A by A1,Th51; then
A17:    h"q in PrimeIdeals(A,E) by A15;
A18:    dom(Spec h) = the carrier of ZariskiTS B by FUNCT_2:def 1;
        q in Spectrum B; then
        reconsider q1 = q as Point of ZariskiTS B by Def7;
        (Spec h).q1 in PrimeIdeals(A,E) by A17,A1,Def9;
        hence thesis by A12,A18,FUNCT_1:def 7;
    end;
