
theorem ThSPrEM1:
  for L being Z_Lattice, r being Element of F_Rat,
  v, u being Vector of EMbedding(r, L) holds
  ((ScProductDM(L)) || (the carrier of EMbedding(r, L))).(v, u)
  = (ScProductDM(L)).(v, u)
  proof
    let L be Z_Lattice, r be Element of F_Rat,
    v, u be Vector of EMbedding(r, L);
    set Z = EMbedding(r, L);
    Z is Submodule of DivisibleMod(L) by ZMODUL08:32;
    then the carrier of Z
    c= the carrier of DivisibleMod(L) by VECTSP_4:def 2;
    then [:the carrier of Z, the carrier of Z:]
    c= [:the carrier of DivisibleMod(L), the carrier of DivisibleMod(L):]
    by ZFMISC_1:96;
    then reconsider sc = (ScProductDM(L)) || (the carrier of Z) as Function of
    [: the carrier of Z, the carrier of Z:], the carrier of F_Real
    by FUNCT_2:32;
    [v, u] in [:the carrier of Z, the carrier of Z:];
    hence thesis by FUNCT_1:49;
  end;
