
theorem
  for L being positive-definite Z_Lattice, b being OrdBasis of EMLat(L),
  v, w being Vector of DivisibleMod(L)
  st for n being Nat st n in dom b holds
  (ScProductDM(L)).(v, b/.n) = (ScProductDM(L)).(w, b/.n)
  holds v = w
  proof
    let L be positive-definite Z_Lattice, b be OrdBasis of EMLat(L),
    v, w be Vector of DivisibleMod(L) such that
    A1: for n being Nat st n in dom b holds
    (ScProductDM(L)).(v, b/.n) = (ScProductDM(L)).(w, b/.n);
    for n being Nat st n in dom b holds
    (ScProductDM(L)).(b/.n, v) = (ScProductDM(L)).(b/.n, w)
    proof
      let n be Nat such that
      B1: n in dom b;
      B2: EMLat(L) is Submodule of DivisibleMod(L) by ThDivisibleL1;
      b/.n in EMLat(L);
      then b/.n in DivisibleMod(L) by B2,ZMODUL01:24;
      then reconsider bn = b/.n as Vector of DivisibleMod(L);
      thus (ScProductDM(L)).(b/.n, v) = (ScProductDM(L)).(v, bn) by ThSPDM1
      .= (ScProductDM(L)).(w, b/.n) by A1,B1
      .= (ScProductDM(L)).(bn, w) by ThSPDM1
      .= (ScProductDM(L)).(b/.n, w);
    end;
    hence thesis by ZL2ThSc1D;
  end;
