
theorem Th9:
  for L be lower-bounded LATTICE holds L
  has_a_representation_of_type<= 2 implies L is modular
proof
  let L be lower-bounded LATTICE;
  assume L has_a_representation_of_type<= 2;
  then consider
  A being non trivial set, f being Homomorphism of L,EqRelLATT A such
  that
A1: f is one-to-one and
A2: ( Image f is finitely_typed & ex e being Equivalence_Relation of A
  st e in the carrier of Image f & e <> id A )& type_of Image f <= 2;
A3: rng (corestr f) = the carrier of Image f & for x,y being Element of L
holds x <= y implies (corestr f).x <= (corestr f).y by FUNCT_2:def 3
,WAYBEL_1:def 2;
  corestr f is one-to-one & for x,y being Element of L holds (corestr f).x
  <= (corestr f).y implies x <= y by A1,Th7,WAYBEL_1:30;
  then corestr f is isomorphic by A3,WAYBEL_0:66;
  then
A4: L, Image f are_isomorphic by WAYBEL_1:def 8;
A5: Image f is lower-bounded LATTICE by Th6;
  Image f is modular by A2,Th8;
  hence thesis by A5,A4,Th5,WAYBEL_1:6;
end;
