
theorem

:: 1.8. THEOREM, (1) <=> (3), p. 145
  for S,T being Lawson complete TopLattice
  for f being SemilatticeHomomorphism of S,T holds
  f is continuous iff f is lim_infs-preserving
proof
  let S,T be Lawson complete TopLattice;
  let f be SemilatticeHomomorphism of S,T;
  thus f is continuous implies f is lim_infs-preserving
  proof
    assume f is continuous;
    then
A1: f is infs-preserving directed-sups-preserving by Th46;
    let N be net of S;
    set M = f*N;
    set Y = the set of all "/\"({M.i where i is Element of M:i >= j},T)
    where j is Element of M;
    reconsider X = the set of all "/\" ({N.i where i is Element of N:i >= j},S)
    where j is Element of N
    as directed non empty Subset of S by Th25;
A2: ex_sup_of X,S by YELLOW_0:17;
A3: f preserves_sup_of X by A1;
A4: the RelStr of f*N = the RelStr of N by WAYBEL_9:def 8;
A5: the carrier of S c= dom f by FUNCT_2:def 1;
    deffunc A(Element of N) = {N.i where i is Element of N: i >= $1};
    deffunc INF(Element of N) = "/\"(A($1),S);
    defpred P[set] means not contradiction;
A6: f.:{INF(i) where i is Element of N: P[i]}
    = {f.INF(i) where i is Element of N: P[i]} from LATTICE3:sch 2(A5);
A7: f.:X = Y
    proof
A8:   now
        let j be Element of N;
        let j9 be Element of M such that
A9:     j9 = j;
        defpred Q[Element of N] means $1 >= j;
        defpred Q9[Element of M] means $1 >= j9;
        deffunc F(Element of N) = f.(N.$1);
        deffunc G(set) = f.((the mapping of N).$1);
A10:    for v being Element of N st Q[v] holds F(v) = G(v);
        deffunc H(set) = (f*the mapping of N).$1;
        deffunc I(Element of M) = M.$1;
A11:    for v being Element of N st Q[v] holds G(v) = H(v) by FUNCT_2:15;
A12:    for v being Element of M st Q9[v] holds H(v) = I(v) by WAYBEL_9:def 8;
        defpred P[set] means [j9,$1] in the InternalRel of N;
A13:    for v being Element of N holds Q[v] iff P[v] by A9,ORDERS_2:def 5;
A14:    for v being Element of M holds P[v] iff Q9[v] by A4,ORDERS_2:def 5;
        deffunc N(Element of N) = N.$1;
        thus f.:A(j) = f.:{N(i) where i is Element of N: Q[i]}
          .= {f.(N(k)) where k is Element of N: Q[k]} from LATTICE3:sch 2(A5)
          .= {F(k) where k is Element of N: Q[k]}
          .= {G(s) where s is Element of N: Q[s]} from FRAENKEL:sch 6(A10)
          .= {H(o) where o is Element of N: Q[o]} from FRAENKEL:sch 6(A11)
          .= {H(r) where r is Element of N: P[r]} from FRAENKEL:sch 3(A13)
          .= {H(m) where m is Element of M: P[m]} by A4
          .= {H(q) where q is Element of M: Q9[q]} from FRAENKEL:sch 3(A14)
          .= {I(n) where n is Element of M: Q9[n]} from FRAENKEL:sch 6(A12)
          .= {M.n where n is Element of M: n >= j9};
      end;
A15:  now
        let j be Element of N;
        A(j) c= the carrier of S
        proof
          let b be object;
          assume b in A(j);
          then ex i being Element of N st b = N.i & i >= j;
          hence thesis;
        end;
        then reconsider A = A(j) as Subset of S;
A16:    f preserves_inf_of A by A1;
        ex_inf_of A,S by YELLOW_0:17;
        hence f."/\"(A(j),S) = "/\"(f.:A(j), T) by A16;
      end;
      thus f.:X c= Y
      proof
        let a be object;
        assume a in f.:X;
        then consider j being Element of N such that
A17:    a = f."/\" ({N.i where i is Element of N:i >= j},S) by A6;
A18:    a = "/\"(f.:A(j),T) by A15,A17;
        reconsider j9 = j as Element of M by A4;
        f.:A(j) = {M.n where n is Element of M: n >= j9} by A8;
        hence thesis by A18;
      end;
      let a be object;
      assume a in Y;
      then consider j9 being Element of M such that
A19:  a = "/\"({M.n where n is Element of M: n >= j9},T);
      reconsider j = j9 as Element of N by A4;
      a = "/\"(f.:A(j),T) by A8,A19
        .= f."/\"(A(j),S) by A15;
      hence thesis by A6;
    end;
    thus f.lim_inf N = f.sup X by WAYBEL11:def 6
      .= sup (f.:X) by A2,A3
      .= lim_inf (f*N) by A7,WAYBEL11:def 6;
  end;
  assume
A20: for N being net of S holds f.lim_inf N = lim_inf (f*N);
A21: f is directed-sups-preserving
  proof
    let D be Subset of S;
    assume D is non empty directed;
    then reconsider D9 = D as non empty directed Subset of S;
    assume ex_sup_of D, S;
    thus ex_sup_of f.:D, T by YELLOW_0:17;
    thus f.sup D = f.lim_inf Net-Str D9 by WAYBEL17:10
      .= lim_inf (f*Net-Str D9) by A20
      .= sup (f.:D) by Th33;
  end;
A22: for X being finite Subset of S holds f preserves_inf_of X by Def1;
  now
    let X be non empty filtered Subset of S;
    reconsider fX = f.:X as filtered non empty Subset of T by WAYBEL20:24;
    thus f preserves_inf_of X
    proof
      assume ex_inf_of X,S;
      thus ex_inf_of f.:X,T by YELLOW_0:17;
      f.inf X = f.lim_inf (X opp+id) by Th28
        .= lim_inf (f*(X opp+id)) by A20
        .= inf fX by Th29
        .= lim_inf (fX opp+id) by Th28;
      hence thesis by Th28;
    end;
  end;
  then f is infs-preserving by A22,WAYBEL_0:71;
  hence thesis by A21,Th46;
end;
