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 ::Theorem 2.7 (ii)
  L is continuous implies for S being CLSubFrame of L holds S is
  continuous LATTICE
proof
  assume
A1: L is continuous;
  let S be CLSubFrame of L;
  reconsider L9= L as complete LATTICE;
A2: S is complete LATTICE by YELLOW_2:30;
  for J, K for F being DoubleIndexedSet of K, S st for j being Element of
  J holds rng(F.j) is directed holds Inf Sups F = Sup Infs Frege F
  proof
    let J, K;
    let F be DoubleIndexedSet of K, S such that
A3: for j being Element of J holds rng(F.j) is directed;
    now
      let j be object;
      assume
   j in J;
      then reconsider j9= j as Element of J;
A5:   F.j9 is Function of K.j9, the carrier of S & the carrier of S c= the
      carrier of L by YELLOW_0:def 13;
      thus F.j is Function of K.j, (J --> the carrier of L).j by A5,FUNCT_2:7;
    end;
    then reconsider F9= F as DoubleIndexedSet of K, L by PBOOLE:def 15;
    ex_inf_of rng Sups F,L by YELLOW_0:17;
    then
A6: "/\"(rng Sups F,L) in the carrier of S by YELLOW_0:def 18;
    now
      let x be object;
      assume x in rng Sups F9;
      then consider j being Element of J such that
A7:   x = Sup(F9.j) by Th14;
A8:   ex_sup_of rng(F.j),L9 & rng(F.j) is directed Subset of S by A3,
YELLOW_0:17;
      x = sup rng(F9.j) by A7,YELLOW_2:def 5
        .= sup rng(F.j) by A8,WAYBEL_0:7
        .= Sup(F.j) by YELLOW_2:def 5;
      hence x in rng Sups F by Th14;
    end;
    then
A9: rng Sups F9 c= rng Sups F;
    now
      let x be object;
      assume x in rng Sups F;
      then consider j being Element of J such that
A10:  x = Sup(F.j) by Th14;
A11:  ex_sup_of rng(F.j),L9 & rng(F.j) is directed Subset of S by A3,
YELLOW_0:17;
      x = sup rng(F.j) by A10,YELLOW_2:def 5
        .= sup rng(F9.j) by A11,WAYBEL_0:7
        .= Sup(F9.j) by YELLOW_2:def 5;
      hence x in rng Sups F9 by Th14;
    end;
    then rng Sups F c= rng Sups F9;
    then ex_inf_of (rng Sups F9),L9 & rng Sups F9 = rng Sups F by A9,
XBOOLE_0:def 10,YELLOW_0:17;
    then inf rng Sups F9 = inf rng Sups F by A6,YELLOW_0:63;
    then
A12: Inf Sups F9 = inf rng Sups F by YELLOW_2:def 6;
    now
      let x be object;
      assume x in rng Infs Frege F;
      then consider f being object such that
A13:  f in dom(Frege F) and
A14:  x = //\((Frege F).f, S) by Th13;
      reconsider f as Function by A13;
      (Frege F).f is Function of J, the carrier of S by A13,Th10;
      then
A15:  rng((Frege F).f) c= the carrier of S by RELAT_1:def 19;
A16:  ex_inf_of rng((Frege F).f),L9 by YELLOW_0:17;
      then
A17:  "/\"(rng((Frege F).f), L) in the carrier of S by A15,YELLOW_0:def 18;
      x = "/\"(rng((Frege F).f), S) by A14,YELLOW_2:def 6
        .= "/\"(rng((Frege F9).f), L) by A15,A16,A17,YELLOW_0:63
        .= //\((Frege F9).f, L) by YELLOW_2:def 6;
      hence x in rng Infs Frege F9 by A13,Th13;
    end;
    then
A18: rng Infs Frege F c= rng Infs Frege F9;
    now
      let x be object;
      assume x in rng /\\(Frege F9, L);
      then consider f being object such that
A19:  f in dom(Frege F9) and
A20:  x = //\((Frege F9).f, L) by Th13;
      reconsider f as Element of product doms F9 by A19;
      (Frege F).f is Function of J, the carrier of S by A19,Th10;
      then
A21:  rng((Frege F).f) c= the carrier of S by RELAT_1:def 19;
A22:  ex_inf_of rng((Frege F).f),L9 by YELLOW_0:17;
      then
A23:  "/\"(rng((Frege F).f), L) in the carrier of S by A21,YELLOW_0:def 18;
      x = "/\"(rng((Frege F9).f), L) by A20,YELLOW_2:def 6
        .= "/\"(rng((Frege F).f), S) by A21,A22,A23,YELLOW_0:63
        .= //\((Frege F).f, S) by YELLOW_2:def 6;
      hence x in rng Infs Frege F by A19,Th13;
    end;
    then rng /\\(Frege F9, L) c= rng /\\(Frege F, S);
    then
A24: rng Infs Frege F9 = rng Infs Frege F by A18,XBOOLE_0:def 10;
    for j being Element of J holds rng(F9.j) is directed
    proof
      let j be Element of J;
      rng(F.j) is directed by A3;
      hence thesis by YELLOW_2:7;
    end;
    then
A25: Inf Sups F9 = Sup Infs Frege F9 by A1,Lm8;
    ex_sup_of rng /\\(Frege F9, L),L9 & rng Infs Frege F is non empty
    directed Subset of S by A2,A3,Th27,YELLOW_0:17;
    then sup rng Infs Frege F9 = sup rng Infs Frege F by A24,WAYBEL_0:7;
    then Sup Infs Frege F9 = sup rng Infs Frege F by YELLOW_2:def 5
      .= Sup Infs Frege F by YELLOW_2:def 5;
    hence thesis by A25,A12,YELLOW_2:def 6;
  end;
  hence thesis by A2,Lm9;
end;
