
theorem
  for S, T being with_suprema Poset, f being Function of S, T for x, y
being Element of S holds f preserves_sup_of {x,y} iff f.(x "\/" y) = f.x "\/" f
  .y
proof
  let S, T be with_suprema Poset, f be Function of S, T, x, y be Element of S;
A1: dom f = the carrier of S by FUNCT_2:def 1;
  hereby
A2: ex_sup_of {x,y},S by YELLOW_0:20;
    assume
A3: f preserves_sup_of {x,y};
    thus f.(x "\/" y) = f.sup {x,y} by YELLOW_0:41
      .= sup (f.:{x,y}) by A3,A2
      .= sup {f.x,f.y} by A1,FUNCT_1:60
      .= f.x "\/" f.y by YELLOW_0:41;
  end;
  assume
A4: f.(x "\/" y) = f.x "\/" f.y;
  assume ex_sup_of {x,y},S;
  f.:{x,y} = {f.x,f.y} by A1,FUNCT_1:60;
  hence ex_sup_of f.:{x,y},T by YELLOW_0:20;
  thus sup (f.:{x,y}) = sup {f.x,f.y} by A1,FUNCT_1:60
    .= f.x "\/" f.y by YELLOW_0:41
    .= f.sup {x,y} by A4,YELLOW_0:41;
end;
