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

theorem Th49:
  for L being non empty Poset holds dom SupMap L = Ids L & rng
  SupMap L is Subset of L
proof
  let L be non empty Poset;
  set P = InclPoset(Ids L);
  thus dom(SupMap L) = the carrier of P by FUNCT_2:def 1
    .= the carrier of RelStr(#Ids L, RelIncl(Ids L)#) by YELLOW_1:def 1
    .= Ids L;
  thus thesis;
end;
