reserve x, y, i for object,
  L for up-complete Semilattice;
reserve L for complete LATTICE,
  a, b, c for Element of L,
  J for non empty set,
  K for non-empty ManySortedSet of J;
reserve J, K, D for non empty set,
  j for Element of J,
  k for Element of K;

theorem ::Theorem 2.3 (1) iff (3)
  L is continuous iff for J, K for F being Function of [:J, K:], the
  carrier of L for X being Subset of L st X = {a where a is Element of L: ex f
  being non-empty ManySortedSet of J st f in Funcs(J, Fin K) & ex G being
DoubleIndexedSet of f, L st (for j, x st x in f.j holds (G.j).x = F.(j, x)) & a
  = Inf Sups G} holds Inf Sups curry F = sup X by Lm13,Lm14;
