
theorem ThScFSDM1:
  for L being Z_Lattice, f being Function of DivisibleMod(L), INT.Ring,
  F being FinSequence of DivisibleMod(L),
  v, u being Vector of DivisibleMod(L), i being Nat
  st i in dom F & u = F.i
  holds (ScFS(v, f, F)).i = (ScProductDM(L)).(v, f.u * u)
  proof
    let L be Z_Lattice, f be Function of DivisibleMod(L), INT.Ring,
        F be FinSequence of DivisibleMod(L),
        v, u be Vector of DivisibleMod(L), i be Nat such that
    A1: i in dom F & u = F.i;
    A2: F/.i = F.i by A1,PARTFUN1:def 6;
    len(ScFS(v, f, F)) = len F by defScFSDM;
    then i in dom(ScFS(v, f, F)) by A1,FINSEQ_3:29;
    hence thesis by A1,A2,defScFSDM;
  end;
