reserve x for set;

theorem
  for L being LATTICE holds
  L is modular iff not ex K being full Sublattice of L st N_5,K are_isomorphic
proof
  let L be LATTICE;
  thus L is modular implies not ex K being full Sublattice of L st N_5,K
  are_isomorphic
  proof
    assume L is modular;
    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 & c"/\"e = c
& d"/\"e = d & b"/\"c = a & b"/\"d = b & c"/\"d = a & b"\/"c = e & c"\/"d = e)
    by Lm2;
    hence thesis by Th9;
  end;
  assume not ex K being full Sublattice of L st N_5,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 & c"/\"e = c & d
  "/\"e = d & b"/\"c = a & b"/\"d = b & c"/\"d = a & b"\/"c = e & c"\/"d = e)
  by Th9;
  hence thesis by Lm2;
end;
