reserve x,y,Y,Z for set,
  L for LATTICE,
  l for Element of L;

theorem Th1:
  for S,T being Semilattice, f being Function of S,T holds f is meet-preserving
  iff for x,y being Element of S holds f.(x "/\" y) = f.x "/\" f.y
proof
  let S,T be Semilattice, f be Function of S,T;
A1: dom f = the carrier of S by FUNCT_2:def 1;
  thus f is meet-preserving implies for x,y being Element of S holds f.(x "/\"
  y) = f. x "/\" f.y
  proof
    assume
A2: f is meet-preserving;
    let z,y be Element of S;
A3: f preserves_inf_of {z,y} by A2;
A4: f.:{z,y} = {f.z,f.y} & ex_inf_of {z,y},S by A1,FUNCT_1:60,YELLOW_0:21;
    thus f.(z "/\" y) = f.inf {z,y} by YELLOW_0:40
      .= inf({f.z,f.y}) by A4,A3
      .= f.z "/\" f.y by YELLOW_0:40;
  end;
  assume
A5: for x,y being Element of S holds f.(x "/\" y) = f. x "/\" f.y;
  for z,y being Element of S holds f preserves_inf_of {z,y}
  proof
    let z,y be Element of S;
A6: f.:{z,y} = {f.z,f.y} by A1,FUNCT_1:60;
    then
A7: ex_inf_of {z,y},S implies ex_inf_of f.:{z,y},T by YELLOW_0:21;
    inf (f.:{z,y}) = f.z "/\" f.y by A6,YELLOW_0:40
      .= f.(z "/\" y) by A5
      .= f.inf {z,y} by YELLOW_0:40;
    hence thesis by A7;
  end;
  hence thesis;
end;
