
theorem Th5:
  for S,T being Semilattice, f being Function of S,T st f is infs-preserving
  holds f is SemilatticeHomomorphism of S,T
proof
  let S,T be Semilattice, f be Function of S,T such that
A1: f is infs-preserving;
  reconsider e = {} as Subset of S by XBOOLE_1:2;
  hereby
    assume S is upper-bounded;
    then
A2: ex_inf_of e, S by YELLOW_0:43;
    f preserves_inf_of e by A1;
    then
A3: ex_inf_of f.:e, T by A2;
    f.:e = {};
    hence T is upper-bounded by A3,WAYBEL20:5;
  end;
  let X be finite Subset of S;
  thus thesis by A1;
end;
