 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
  h is RingHomomorphism implies Spec h is continuous
    proof
      assume
A1:   h is RingHomomorphism;
      for P1 being Subset of ZariskiTS A st P1 is closed holds
        (Spec h)" P1 is closed
      proof
        let P1 be Subset of ZariskiTS A;
        assume P1 is closed; then
        consider E be non empty Subset of A such that
A3:     P1 = PrimeIdeals(A,E) by Def7;
A4:     dom h = the carrier of A by FUNCT_2:def 1;
        (Spec h)"P1 = PrimeIdeals(B,h.:E) by A3,A1,Th52; then
        consider E1 be non empty Subset of B such that
        E1 = h.:E and
A5:     (Spec h)"P1 = PrimeIdeals(B,E1) by A4;
        thus thesis by A5,Def7;
      end;
      hence thesis by PRE_TOPC:def 6;
    end;
