
theorem
  for L be lower-bounded LATTICE holds ( L is algebraic implies ex X be
  non empty set, c be closure Function of BoolePoset X,BoolePoset X st c is
  directed-sups-preserving & L,Image c are_isomorphic ) & (( ex X be set, c be
closure Function of BoolePoset X,BoolePoset X st c is directed-sups-preserving
  & L,Image c are_isomorphic ) implies L is algebraic )
proof
  let L be lower-bounded LATTICE;
  hereby
    assume L is algebraic;
    then 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;
    hence ex X be non empty set, c be closure Function of BoolePoset X,
    BoolePoset X st c is directed-sups-preserving & L,Image c are_isomorphic
by Lm2;
  end;
  thus thesis by Lm5;
end;
