
theorem
  for L being positive-definite finite-rank free Z_Lattice holds
  GramDet(InnerProduct(L)) = GramDet(InnerProduct(EMLat(L)))
  proof
    let L be positive-definite finite-rank free Z_Lattice;
    set b = the OrdBasis of L;
    reconsider e = (MorphsZQ(L))*b as OrdBasis of EMLat(L) by ZMODLAT2:41;
    P1: GramMatrix(InnerProduct(L), b) = GramMatrix(InnerProduct(EMLat(L)), e)
    by LmEMDetX3;
    GramDet(InnerProduct(L)) = Det GramMatrix(InnerProduct(L), b)
    by ZMODLAT1:def 35
    .= Det GramMatrix(InnerProduct(EMLat(L)), e) by P1,ZMODLAT2:42
    .= GramDet(InnerProduct(EMLat(L))) by ZMODLAT1:def 35;
    hence thesis;
  end;
