
theorem
  for S,T being upper-bounded Semilattice, f being meet-preserving Function
  of S, T st f.Top S = Top T holds f is SemilatticeHomomorphism of S,T
proof
  let S,T be upper-bounded Semilattice, f be meet-preserving Function of S,T
  such that
A1: f.Top S = Top T;
  thus S is upper-bounded implies T is upper-bounded;
  let X be finite Subset of S;
A2: ex_inf_of f.:{}, T by YELLOW_0:43;
  X = {} or f preserves_inf_of X by Th2;
  hence thesis by A1,A2;
end;
