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 Th74:
  for V be RealLinearSpace,
     Fr be FinSequence of V,
     fr be Function of V,REAL,
     Fv be FinSequence of RLSp2RVSp(V),
     fv be Function of RLSp2RVSp(V),F_Real
    st fr = fv & Fr = Fv
  holds fr(#)Fr = fv(#)Fv
  proof
    let V be RealLinearSpace,
       Fr be FinSequence of V,
       fr be Function of V,REAL,
       Fv be FinSequence of RLSp2RVSp(V),
       fv be Function of RLSp2RVSp(V),F_Real;

    assume
    A1: fr = fv & Fr = Fv; then

    A2: len (fv(#)Fv) = len Fr by VECTSP_6:def 5;

    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 i in dom (fv(#)Fv);
      hence (fv(#)Fv) . i
       = (fv . (Fv /. i)) * (Fv /. i) by VECTSP_6:def 5
      .= (fr . (Fr /. i)) * (Fr /. i) by A1;
    end;
    hence thesis by A2,RLVECT_2:def 7;
  end;
