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 Th3:
  for F be FinSequence of TOP-REAL n,
      fr be Function of TOP-REAL n,REAL,
      Fv be FinSequence of n-VectSp_over F_Real,
      fv be Function of n-VectSp_over F_Real,F_Real st fr = fv & F = Fv
  holds fr(#)F = fv(#)Fv
proof
  let F be FinSequence of TOP-REAL n,
   fr be Function of TOP-REAL n,REAL,
   Fv be FinSequence of n-VectSp_over F_Real,
   fv be Function of n-VectSp_over F_Real,F_Real;
  assume that
   A1: fr=fv and
   A2: F=Fv;
  A3: len(fv(#)Fv)=len Fv by VECTSP_6:def 5;
  A4: len(fr(#)F)=len F by RLVECT_2:def 7;
  now reconsider T=TOP-REAL n as RealLinearSpace;
   let i be Nat;
   reconsider Fi=F/.i as FinSequence of REAL by EUCLID:24;
   reconsider Fvi=Fv/.i as Element of n-tuples_on the carrier of F_Real
     by MATRIX13:102;
   reconsider Fii=F/.i as Element of T;
   assume A5: 1<=i & i<=len F;
   then A6: i in dom(fv(#)Fv) by A2,A3,FINSEQ_3:25;
   i in dom F by A5,FINSEQ_3:25;
   then A7: F/.i=F.i by PARTFUN1:def 6;
   i in dom Fv by A2,A5,FINSEQ_3:25;
   then A8: Fv/.i=Fv.i by PARTFUN1:def 6;
   i in dom(fr(#)F) by A4,A5,FINSEQ_3:25;
   hence (fr(#)F).i=fr.Fii*Fii by RLVECT_2:def 7
    .=fr.Fi*Fi by EUCLID:65
    .=fv.(Fv/.i)*Fvi by A1,A2,A7,A8,MATRIXR1:17
    .=fv.(Fv/.i)*(Fv/.i) by MATRIX13:102
    .=(fv(#)Fv).i by A6,VECTSP_6:def 5;
  end;
  hence thesis by A2,A4,A3;
end;
