reserve X for set,
  F for Filter of BoolePoset X,
  x for Element of BoolePoset X ,
  z for Element of X;

theorem Th17:
  for L, M being continuous complete LATTICE, F, G being set st F
  is_FreeGen_set_of L & G is_FreeGen_set_of M & card F = card G holds L, M
  are_isomorphic
proof
  let L, M be continuous complete LATTICE, Lg, Mg be set such that
A1: Lg is_FreeGen_set_of L and
A2: Mg is_FreeGen_set_of M and
A3: card Lg = card Mg;
  Lg,Mg are_equipotent by A3,CARD_1:5;
  then consider f being Function such that
A4: f is one-to-one and
A5: dom f = Lg and
A6: rng f = Mg;
  set g = f";
A7: dom g = Mg by A4,A6,FUNCT_1:33;
  reconsider Mg as Subset of M by A2,Th7;
A8: rng g = Lg by A4,A5,FUNCT_1:33;
  reconsider Lg as Subset of L by A1,Th7;
  Mg c= the carrier of M;
  then reconsider f as Function of Lg, the carrier of M by A5,A6,FUNCT_2:def 1
,RELSET_1:4;
  consider F being CLHomomorphism of L, M such that
A9: F|Lg = f and
  for h9 being CLHomomorphism of L, M st h9|Lg = f holds h9 = F by A1;
  Lg c= the carrier of L;
  then reconsider g as Function of Mg, the carrier of L by A7,A8,FUNCT_2:def 1
,RELSET_1:4;
  consider G being CLHomomorphism of M, L such that
A10: G|Mg = g and
  for h9 being CLHomomorphism of M, L st h9|Mg = g holds h9 = G by A2;
  reconsider GF = G*F as CLHomomorphism of L, L by Th2;
  GF|Lg = G*f by A9,RELAT_1:83
    .= g*f by A6,A10,FUNCT_4:2
    .= id Lg by A4,A5,FUNCT_1:39;
  then
A11: GF = id L by A1,Th8;
  then
A12: F is one-to-one by FUNCT_2:23;
  reconsider FG = F*G as CLHomomorphism of M, M by Th2;
  F is directed-sups-preserving by WAYBEL16:def 1;
  then
A13: F is monotone by WAYBEL17:3;
  G is directed-sups-preserving by WAYBEL16:def 1;
  then
A14: G is monotone by WAYBEL17:3;
  FG|Mg = F*g by A10,RELAT_1:83
    .= f*g by A8,A9,FUNCT_4:2
    .= id Mg by A4,A6,FUNCT_1:39;
  then FG = id M by A2,Th8;
  then F is onto by FUNCT_2:23;
  then rng F = the carrier of M by FUNCT_2:def 3;
  then G = (F qua Function)" by A11,A12,FUNCT_2:30;
  then F is isomorphic by A12,A13,A14,WAYBEL_0:def 38;
  hence thesis by WAYBEL_1:def 8;
end;
