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;
