
theorem ThScFS3:
  for L being Z_Lattice, f being Function of L, INT.Ring,
      v, u being Vector of L holds
  ScFS(v, f, <* u *>) = <* <; v, f.u * u ;> *>
  proof
    let L be Z_Lattice, f be Function of L, INT.Ring;
    let v, u be Vector of L;
    A1: 1 in {1} by TARSKI:def 1;
    A2: len ScFS(v, f, <* u *>) = len <* u *> by defScFS
    .= 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 = <; v, f.(<* u *>/.1) * <* u *>/.1 ;>
    by A1,defScFS
    .= <; v, f.(<* u *>/.1) * u ;> by FINSEQ_4:16
    .= <; v, f.u * u ;> by FINSEQ_4:16;
    hence thesis by A2,FINSEQ_1:40;
  end;
