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

theorem Th2:
  for S,T being sup-Semilattice, f being Function of S,T holds f is
join-preserving iff for x,y being Element of S holds f.(x "\/" y) = f. x "\/" f
  .y
proof
  let S,T be sup-Semilattice, f be Function of S,T;
A1: dom f = the carrier of S by FUNCT_2:def 1;
  thus f is join-preserving implies for x,y being Element of S holds f.(x "\/"
  y) = f. x "\/" f.y
  proof
    assume
A2: f is join-preserving;
    let z,y be Element of S;
A3: f preserves_sup_of {z,y} by A2;
A4: f.:{z,y} = {f.z,f.y} & ex_sup_of {z,y},S by A1,FUNCT_1:60,YELLOW_0:20;
    thus f.(z "\/" y) = f.sup {z,y} by YELLOW_0:41
      .= sup ({f.z,f.y}) by A4,A3
      .= f.z "\/" f.y by YELLOW_0:41;
  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_sup_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_sup_of {z,y},S implies ex_sup_of f.:{z,y},T by YELLOW_0:20;
    sup (f.:{z,y}) = f.z "\/" f.y by A6,YELLOW_0:41
      .= f.(z "\/" y) by A5
      .= f.sup {z,y} by YELLOW_0:41;
    hence thesis by A7;
  end;
  hence thesis;
end;
