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 Th22:
  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
proof
  let F be Function of [:J, K:], the carrier of L;
  let X be Subset of L;
A1: for f being non-empty ManySortedSet of J st f in Funcs(J, Fin K) for G
being DoubleIndexedSet of f, L st for j, x st x in f.j holds (G.j).x = F.(j, x)
  for j holds Sup((curry F).j) >= Sup(G.j)
  proof
    let f be non-empty ManySortedSet of J such that
A2: f in Funcs(J, Fin K);
    let G be DoubleIndexedSet of f, L such that
A3: for j, x st x in f.j holds (G.j).x = F.(j, x);
    let j;
A4: ex_sup_of rng((curry F).j), L & ex_sup_of rng(G.j),L by YELLOW_0:17;
    rng(G.j) is Subset of rng((curry F).j) by A2,A3,Lm12;
    then sup rng((curry F).j) >= sup rng(G.j) by A4,YELLOW_0:34;
    then Sup((curry F).j) >= sup rng(G.j) by YELLOW_2:def 5;
    hence thesis by YELLOW_2:def 5;
  end;
A5: for f being non-empty ManySortedSet of J st f in Funcs(J, Fin K) for G
being DoubleIndexedSet of f, L st for j, x st x in f.j holds (G.j).x = F.(j, x)
  holds Inf Sups curry F >= Inf Sups G
  proof
    let f be non-empty ManySortedSet of J such that
A6: f in Funcs(J, Fin K);
    let G be DoubleIndexedSet of f, L such that
A7: for j, x st x in f.j holds (G.j).x = F.(j, x);
    rng Sups curry F is_>=_than Inf Sups G
    proof
      let a;
      assume a in rng Sups curry F;
      then consider j being Element of J such that
A8:   a = Sup((curry F).j) by Th14;
      Sup(G.j) in rng Sups G by Th14;
      then Sup(G.j) >= inf rng Sups G by YELLOW_2:22;
      then
A9:   Sup(G.j) >= Inf Sups G by YELLOW_2:def 6;
      Sup((curry F).j) >= Sup(G.j) by A1,A6,A7;
      hence thesis by A8,A9,ORDERS_2:3;
    end;
    then inf rng Sups curry F >= Inf Sups G by YELLOW_0:33;
    hence thesis by YELLOW_2:def 6;
  end;
  assume
A10: 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};
  Inf Sups curry F is_>=_than X
  proof
    let a;
    assume a in X;
    then ex a9 being Element of L st a9 = a & 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)) & a9 = Inf Sups G by A10;
    hence thesis by A5;
  end;
  hence thesis by YELLOW_0:32;
end;
