
theorem Th58:
  for L be up-complete non empty Poset for S be non empty full
  SubRelStr of L holds supMap S is monotone
proof
  let L be up-complete non empty Poset;
  let S be non empty full SubRelStr of L;
  set f = supMap S;
  now
    let x, y be Element of InclPoset Ids S;
    reconsider I = x, J = y as Ideal of S by YELLOW_2:41;
    assume x <= y;
    then
A1: I c= J by YELLOW_1:3;
    I is non empty directed Subset of L by YELLOW_2:7;
    then
A2: ex_sup_of I,L by WAYBEL_0:75;
    J is non empty directed Subset of L by YELLOW_2:7;
    then
A3: ex_sup_of J,L by WAYBEL_0:75;
A4: f.y = "\/"(J,L) by Def10;
    f.x = "\/"(I,L) by Def10;
    hence f.x <= f.y by A2,A3,A1,A4,YELLOW_0:34;
  end;
  hence thesis by WAYBEL_1:def 2;
end;
