
theorem
  for L being RATional positive-definite Z_Lattice,
  b being OrdBasis of EMLat(L), I being Basis of EMLat(L)
  st I = rng b holds
  (B2DB(I))*b is OrdBasis of DualLat(L)
  proof
    let L be RATional positive-definite Z_Lattice,
    b be OrdBasis of EMLat(L), I be Basis of EMLat(L) such that
    A1: I = rng b;
    A2: b is one-to-one by ZMATRLIN:def 5;
    b is FinSequence of I by A1,FINSEQ_1:def 4; then
    A3: (B2DB(I))*b is FinSequence of DualBasis(I) by FINSEQ_2:32;
    A4: dom(B2DB(I)) = I by defB2DB;
    A5: rng((B2DB(I))*b) = (B2DB(I)).: (rng b) by RELAT_1:127
    .= rng(B2DB(I)) by A1,A4,RELAT_1:113
    .= DualBasis(I) by defB2DB;
    DualBasis(I) is Subset of DualLat(L) by ThDLDB;
    then A6: (B2DB(I))*b is FinSequence of DualLat(L) by A3,A5,FINSEQ_1:def 4;
    rng((B2DB(I))*b) is Basis of DualLat(L) by A5,ThDLDB;
    hence thesis by A2,A6,ZMATRLIN:def 5;
  end;
