
theorem LmEMDetX9:
  for L being Z_Lattice,
  E being finite-rank free Z_Module,
  I being FinSequence of L,
  J being FinSequence of E
  st the ModuleStr of L = the ModuleStr of E & I = J
  holds ( I is OrdBasis of L iff J is OrdBasis of E)
  proof
    let L be Z_Lattice, E be finite-rank free Z_Module,
    I be FinSequence of L, J be FinSequence of E;
    assume that
    AS1: the ModuleStr of L = the ModuleStr of E and
    AS2: I = J;
    hereby
      assume I is OrdBasis of L;
      then P2: I is one-to-one & rng I is Basis of L by ZMATRLIN:def 5;
      then rng J is Basis of E by AS1,AS2,LmEMDetX5;
      hence J is OrdBasis of E by AS2,P2,ZMATRLIN:def 5;
    end;
    assume J is OrdBasis of E;
    then P2: J is one-to-one & rng J is Basis of E by ZMATRLIN:def 5;
    then rng I is Basis of L by AS1,AS2,LmEMDetX5;
    hence I is OrdBasis of L by AS2,P2,ZMATRLIN:def 5;
  end;
