
theorem
  for L be lower-bounded LATTICE holds ( L is algebraic implies ex X be
non empty set, S be full SubRelStr of BoolePoset X st S is infs-inheriting & S
is directed-sups-inheriting & L,S are_isomorphic ) & (( ex X be set, S be full
  SubRelStr of BoolePoset X st S is infs-inheriting & S is
  directed-sups-inheriting & L,S are_isomorphic ) implies L is algebraic )
proof
  let L be lower-bounded LATTICE;
  thus L is algebraic implies ex X be non empty set, S be full SubRelStr of
  BoolePoset X st S is infs-inheriting & S is directed-sups-inheriting & L,S
  are_isomorphic by Lm1;
  thus (ex X be set, S be full SubRelStr of BoolePoset X st ( S is
infs-inheriting & S is directed-sups-inheriting & L,S are_isomorphic )) implies
  L is algebraic
  proof
    assume ex X be set, S be full SubRelStr of BoolePoset X st S is
    infs-inheriting & S is directed-sups-inheriting & L,S are_isomorphic;
    then ex X be set, c be closure Function of BoolePoset X,BoolePoset X st c
    is directed-sups-preserving & L,Image c are_isomorphic by Lm3;
    hence thesis by Lm5;
  end;
end;
