
theorem
  for L being continuous LATTICE, x,y being Element of L
  holds x <= y iff waybelow x c= waybelow y
proof
  let L be continuous LATTICE, x,y be Element of L;
  thus x <= y implies waybelow x c= waybelow y by Th12;
  assume
A1: waybelow x c= waybelow y;
A2: ex_sup_of waybelow x,L by WAYBEL_0:75;
  ex_sup_of waybelow y,L by WAYBEL_0:75;
  then sup waybelow x <= sup waybelow y by A1,A2,YELLOW_0:34;
  then x <= sup waybelow y by Def5;
  hence thesis by Def5;
end;
