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;
reserve J for non empty set,
  K for non-empty ManySortedSet of J;

theorem Th27:
  for F being DoubleIndexedSet of K, L st for j being Element of J
  holds rng(F.j) is directed holds rng Infs Frege F is directed
proof
  let F be DoubleIndexedSet of K, L;
  set X = rng Infs Frege F;
  assume
A1: for j being Element of J holds rng(F.j) is directed;
  for x, y being Element of L st x in X & y in X ex z being Element of L
  st z in X & x <= z & y <= z
  proof
    let x, y be Element of L;
    assume that
A2: x in X and
A3: y in X;
    consider h being object such that
A4: h in dom(Frege F) and
A5: y = //\((Frege F).h, L) by A3,Th13;
    reconsider h as Function by A4;
    reconsider H = (Frege F).h as Function of J, the carrier of L by A4,Th10;
    consider g being object such that
A6: g in dom(Frege F) and
A7: x = //\((Frege F).g, L) by A2,Th13;
    reconsider g as Function by A6;
    reconsider G = (Frege F).g as Function of J, the carrier of L by A6,Th10;
A8: ex_inf_of rng((Frege F).g),L by YELLOW_0:17;
    defpred P[object, object] means
$1 in J & $2 in K.$1 & for c being Element of L
st c = (F.$1).$2 holds (for a being Element of L st a = G.$1 holds a <= c) & (
    for b being Element of L st b = H.$1 holds b <= c);
A9: for j being Element of J ex k being Element of K.j st G.j <= (F.j).k
    & H.j <= (F.j).k
    proof
      let j be Element of J;
      j in J;
      then
A10:  j in dom F by PARTFUN1:def 2;
      then
A11:  g.j in dom(F.j) by A6,Th9;
      j in J;
      then
A12:  j in dom F by PARTFUN1:def 2;
      then G.j = (F.j).(g.j) by A6,Th9;
      then
A13:  G.j in rng(F.j) by A11,FUNCT_1:def 3;
A14:  h.j in dom(F.j) by A4,A10,Th9;
      H.j = (F.j).(h.j) by A4,A12,Th9;
      then
A15:  H.j in rng(F.j) by A14,FUNCT_1:def 3;
      rng(F.j) is directed Subset of L by A1;
      then consider c being Element of L such that
A16:  c in rng(F.j) and
A17:  G.j <= c & H.j <= c by A13,A15,WAYBEL_0:def 1;
      consider k being object such that
A18:  k in dom(F.j) and
A19:  c = (F.j).k by A16,FUNCT_1:def 3;
      reconsider k as Element of K.j by A18;
      take k;
      thus thesis by A17,A19;
    end;
A20: for j being object st j in J
ex k being object st k in union rng K & P[j, k ]
    proof
      let j9 be object;
      assume j9 in J;
      then reconsider j = j9 as Element of J;
      consider k being Element of K.j such that
A21:  G.j <= (F.j).k & H.j <= (F.j).k by A9;
      take k;
      j in J;
      then j in dom K by PARTFUN1:def 2;
      then K.j in rng K by FUNCT_1:def 3;
      hence k in union rng K by TARSKI:def 4;
      thus thesis by A21;
    end;
    consider f being Function such that
A22: dom f = J and
    rng f c= union rng K and
A23: for j being object st j in J holds P[j, f.j] from FUNCT_1:sch 6(A20
    );
A24: now
      let x be object;
      assume x in dom doms F;
      then
A25:  x in dom F by FUNCT_6:59;
      then reconsider j = x as Element of J;
      (doms F).x = dom(F.j) by A25,FUNCT_6:22
        .= K.j by FUNCT_2:def 1;
      hence f.x in (doms F).x by A23;
    end;
    dom f = dom F by A22,PARTFUN1:def 2
      .= dom doms F by FUNCT_6:59;
    then f in product doms F by A24,CARD_3:9;
    then
A26: f in dom(Frege F) by PARTFUN1:def 2;
    then reconsider Ff = (Frege F).f as Function of J, the carrier of L by Th10
;
    take z = Inf Ff;
    thus z in X by A26,Th13;
A27: x = "/\"(rng((Frege F).g), L) by A7,YELLOW_2:def 6;
    now
      let j be Element of J;
A28:  j in J;
      then j in dom G by FUNCT_2:def 1;
      then
A29:  G.j in rng G by FUNCT_1:def 3;
      j in dom F by A28,PARTFUN1:def 2;
      then (F.j).(f.j) = ((Frege F).f).j by A26,Th9;
      then
A30:  G.j <= Ff.j by A23;
      x is_<=_than rng G by A27,A8,YELLOW_0:def 10;
      then x <= G.j by A29;
      hence x <= Ff.j by A30,ORDERS_2:3;
    end;
    hence x <= z by YELLOW_2:55;
A31: ex_inf_of rng((Frege F).h),L by YELLOW_0:17;
A32: y = "/\"(rng((Frege F).h), L) by A5,YELLOW_2:def 6;
    now
      let j be Element of J;
A33:  j in J;
      then j in dom H by FUNCT_2:def 1;
      then
A34:  H.j in rng H by FUNCT_1:def 3;
      j in dom F by A33,PARTFUN1:def 2;
      then (F.j).(f.j) = ((Frege F).f).j by A26,Th9;
      then
A35:  H.j <= Ff.j by A23;
      y is_<=_than rng H by A32,A31,YELLOW_0:def 10;
      then y <= H.j by A34;
      hence y <= Ff.j by A35,ORDERS_2:3;
    end;
    hence thesis by YELLOW_2:55;
  end;
  hence thesis by WAYBEL_0:def 1;
end;
