
theorem ThScFS1:
  for L being Z_Lattice, f being Function of L, INT.Ring,
      F being FinSequence of L,
      v, u being Vector of L, i being Nat
  st i in dom F & u = F.i
  holds (ScFS(v, f, F)).i = <; v, f.u * u ;>
  proof
    let L be Z_Lattice, f be Function of L, INT.Ring,
        F be FinSequence of L,
        v, u be Vector of 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 defScFS;
    then i in dom(ScFS(v, f, F)) by A1,FINSEQ_3:29;
    hence thesis by A1,A2,defScFS;
  end;
