reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem Th19:
  for Fr be FinSequence of TOP-REAL n,
      fr be Function of TOP-REAL n,REAL,
      Fv be FinSequence of REAL-NS n,
      fv be Function of REAL-NS n,REAL
    st fr = fv & Fr = Fv
  holds fr(#)Fr = fv(#)Fv
  proof
    let Fr be FinSequence of TOP-REAL n,
        fr be Function of TOP-REAL n,REAL,
        Fv be FinSequence of REAL-NS n,
        fv be Function of REAL-NS n,REAL;
    assume
    A1: fr = fv & Fr = Fv; then
    A2: len(fv(#)Fv) = len Fr by RLVECT_2:def 7;
    A3: fv(#)Fv is FinSequence of TOP-REAL n by Th4;
    for i be Nat st i in dom (fv(#)Fv) holds
    (fv(#)Fv).i = (fr.(Fr /. i)) * (Fr /. i)
    proof
      let i be Nat;
      assume
      A4: i in dom (fv(#)Fv);
      A5: Fv /. i = Fr /. i by A1,Th4;
      thus (fv(#)Fv).i = (fv . (Fv /. i)) * (Fv /. i) by A4,RLVECT_2:def 7
      .= (fr . (Fr /. i)) * (Fr /. i) by A1,A5,Th8;
    end;
    hence thesis by A2,A3,RLVECT_2:def 7;
  end;
