reserve x, y, i for object,
  L for up-complete Semilattice;

theorem ::Theorem 2.1 (5) implies (4)
  for L being complete Semilattice holds SupMap L is infs-preserving
  sups-preserving implies SupMap L is upper_adjoint
proof
  let L be complete Semilattice;
  set r = SupMap L;
  assume r is infs-preserving sups-preserving;
  then ex d being Function of L, InclPoset(Ids L) st [r, d] is Galois & for t
  being Element of L holds d.t is_minimum_of r"(uparrow t) by WAYBEL_1:14;
  hence thesis by WAYBEL_1:def 11;
end;
