
theorem ThScFSDM3:
  for L being Z_Lattice, f being Function of DivisibleMod(L), INT.Ring,
      v, u being Vector of DivisibleMod(L) holds
  ScFS(v, f, <* u *>) = <* (ScProductDM(L)).(v, f.u * u) *>
  proof
    let L be Z_Lattice, f be Function of DivisibleMod(L), INT.Ring;
    let v, u be Vector of DivisibleMod(L);
    A1: 1 in {1} by TARSKI:def 1;
    A2: len ScFS(v, f, <* u *>) = len <* u *> by defScFSDM
    .= 1 by FINSEQ_1:40;
    then dom(ScFS(v, f, <* u *>)) = {1} by FINSEQ_1:2,FINSEQ_1:def 3;
    then (ScFS(v, f, <* u *>)).1
    = (ScProductDM(L)).(v, f.(<* u *>/.1) * <* u *>/.1) by A1,defScFSDM
    .= (ScProductDM(L)).(v, f.(<* u *>/.1) * u) by FINSEQ_4:16
    .= (ScProductDM(L)).(v, f.u * u) by FINSEQ_4:16;
    hence thesis by A2,FINSEQ_1:40;
  end;
