reserve x for set;

theorem
  for L being LATTICE st L is modular holds L is distributive iff not ex
  K being full Sublattice of L st M_3,K are_isomorphic
proof
  let L be LATTICE;
  assume
A1: L is modular;
  thus L is distributive implies not ex K being full Sublattice of L st M_3,K
  are_isomorphic
  proof
    assume L is distributive;
    then
    not ex a,b,c,d,e being Element of L st (a,b,c,d,e are_mutually_distinct &
    a"/\"b = a & a"/\"c = a & a"/\"d = a
 & b"/\"e = b & c"/\"e = c & d"/\" e = d & b"/\"c = a & b"/\"d = a & c"/\"d = a
    & b"\/"c = e & b"\/"d = e & c"\/" d = e) by Lm3;
    hence thesis by Th10;
  end;
  assume not ex K being full Sublattice of L st M_3,K are_isomorphic;
  then
  not ex a,b,c,d,e being Element of L st (a,b,c,d,e are_mutually_distinct
  & a"/\"b = a & a"/\"c = a & a"/\"d = a & b
"/\"e = b & c"/\"e = c & d"/\" e = d & b"/\"c = a & b"/\"d = a & c"/\"d = a & b
  "\/"c = e & b"\/"d = e & c"\/" d = e) by Th10;
  hence thesis by A1,Lm3;
end;
