
theorem Th47:
  for L being complete LATTICE for J being non empty set,
      K being non-empty ManySortedSet of J
  for F being DoubleIndexedSet of K,L holds
    Sup Infs F <= Inf Sups Frege F
proof
  let L be complete LATTICE;
  let J be non empty set, K be non-empty ManySortedSet of J;
  let F be DoubleIndexedSet of K, L;
  Inf Sups Frege F is_>=_than rng Infs F
  proof
    let x be Element of L;
    assume x in rng Infs F;
    then consider a being Element of J such that
A1: x = Inf (F.a) by WAYBEL_5:14;
A2: x = inf rng (F.a) by A1,YELLOW_2:def 6;
    x is_<=_than rng Sups Frege F
    proof
      reconsider J9 = product doms F as non empty set;
      let y be Element of L;
      reconsider K9 = J9 --> J as ManySortedSet of J9;
      reconsider G = Frege F as DoubleIndexedSet of K9, L;
      assume y in rng Sups Frege F;
      then consider f being Element of J9 such that
A3:   y = Sup (G.f) by WAYBEL_5:14;
      reconsider f as Element of product doms F;
A4:   dom F = J & dom Frege F = product doms F by PARTFUN1:def 2;
      then f.a in dom (F.a) by WAYBEL_5:9;
      then reconsider j = f.a as Element of K.a;
A5:   (F.a).j in rng ((Frege F).f) by A4,WAYBEL_5:9;
      j in dom (F.a) by A4,WAYBEL_5:9;
      then (F.a).j in rng (F.a) by FUNCT_1:def 3;
      then
A6:   x <= (F.a).j by A2,YELLOW_2:22;
      y = sup rng ((Frege F).f) by A3,YELLOW_2:def 5;
      then (F.a).j <= y by A5,YELLOW_2:22;
      hence x <= y by A6,ORDERS_2:3;
    end;
    then x <= inf rng Sups Frege F by YELLOW_0:33;
    hence thesis by YELLOW_2:def 6;
  end;
  then sup rng Infs F <= Inf Sups Frege F by YELLOW_0:32;
  hence thesis by YELLOW_2:def 5;
end;
