reserve x, X, Y for set;
reserve L for complete LATTICE,
  a for Element of L;

theorem Th52:
  for L being non empty Poset holds IdsMap L is monotone
proof
  let L be non empty Poset;
  let x1,x2 be Element of L;
  assume x1 <= x2;
  then downarrow x1 c= downarrow x2 by WAYBEL_0:21;
  then (IdsMap L).x1 c= downarrow x2 by Def4;
  then
A1: (IdsMap L).x1 c= (IdsMap L).x2 by Def4;
  let a, b be Element of InclPoset(Ids L);
  assume a = (IdsMap L).x1 & b = (IdsMap L).x2;
  hence a <= b by A1,YELLOW_1:3;
end;
