
theorem Th10:
  for T being up-complete non empty Poset
  for S being directed-sups-inheriting full non empty SubRelStr of T
  holds incl(S,T) is directed-sups-preserving
proof
  let T be up-complete non empty Poset;
  let S be directed-sups-inheriting full non empty SubRelStr of T;
  set f = incl(S,T);
  let X be Subset of S;
  assume that
A1: X is non empty directed and ex_sup_of X, S;
  reconsider X9 = X as non empty directed Subset of T by A1,YELLOW_2:7;
  the carrier of S c= the carrier of T by YELLOW_0:def 13;
  then
A2: f = id the carrier of S by YELLOW_9:def 1;
  then
A3: f.:X = X9 by FUNCT_1:92;
A4: f.sup X = sup X by A2;
  thus ex_sup_of f.:X, T by A3,WAYBEL_0:75;
  hence thesis by A1,A3,A4,WAYBEL_0:7;
end;
