
theorem Th46:

:: 1.8. THEOREM, (1) <=> (2), p. 145
  for S,T being Lawson complete TopLattice
  for f being SemilatticeHomomorphism of S,T holds
  f is continuous iff f is infs-preserving directed-sups-preserving
proof
  let S,T be Lawson complete TopLattice;
  let f be SemilatticeHomomorphism of S,T;
  hereby
    assume
A1: f is continuous;
A2: for X being finite Subset of S holds f preserves_inf_of X by Def1;
    for X being non empty filtered Subset of S holds f preserves_inf_of X
    by A1,Th45;
    hence f is infs-preserving by A2,WAYBEL_0:71;
    thus f is directed-sups-preserving by A1,Th45;
  end;
  assume f is infs-preserving;
  then for X being non empty Subset of S holds f preserves_inf_of X;
  hence thesis by Th45;
end;
