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

theorem Th51:
  for L being up-complete non empty Poset holds SupMap L is monotone
proof
  let L be up-complete non empty Poset;
  set P = InclPoset Ids L;
  set f = SupMap L;
  for x, y being Element of P st x <= y for a, b being Element of L st a =
  f.x & b = f.y holds a <= b
  proof
    let x, y be Element of P such that
A1: x <= y;
    reconsider I = x, J = y as Ideal of L by Th41;
A2: I c= J by A1,YELLOW_1:3;
A3: ex_sup_of I,L & ex_sup_of J,L by WAYBEL_0:75;
A4: f.x = sup I & f.y = sup J by Def3;
    let a, b be Element of L;
    assume a = f.x & b = f.y;
    hence thesis by A3,A2,A4,YELLOW_0:34;
  end;
  hence thesis;
end;
