reserve T1,T2,T3 for TopSpace,
  A1 for Subset of T1, A2 for Subset of T2, A3 for Subset of T3;
reserve n,k for Nat;
reserve M,N for non empty TopSpace;
reserve p,q,p1,p2 for Point of TOP-REAL n;
reserve r for Real;

theorem Th18:
  for F be FinSequence of TOP-REAL n,
      fr be Function of TOP-REAL n,REAL,
      Fv be FinSequence of RealVectSpace(Seg n),
      fv be Function of RealVectSpace(Seg n),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 RealVectSpace(Seg n),
   fv be Function of RealVectSpace(Seg n),REAL;
  assume that
A1: fr=fv and
A2: F=Fv;
A3: len(fv(#)Fv)=len Fv by RLVECT_2:def 7;
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;
A5: the carrier of n-VectSp_over F_Real=the carrier of TOP-REAL n
   proof
     thus the carrier of n-VectSp_over F_Real=REAL n by MATRIX13:102
     .=the carrier of TOP-REAL n by EUCLID:22;
   end;
   the carrier of n -VectSp_over F_Real
   = n-tuples_on the carrier of F_Real by MATRIX13:102;
   then reconsider Fvi=Fv/.i as Element of n-tuples_on the carrier of F_Real
     by Lm1,A5;
   reconsider Fii=F/.i as Element of T;
   assume
A6: 1<=i & i<=len F;
   then
A7: i in dom(fv(#)Fv) by A2,A3,FINSEQ_3:25;
   i in dom F by A6,FINSEQ_3:25;
   then
A8: F/.i=F.i by PARTFUN1:def 6;
A9: Fv/.i=Fv.i by A2,A6,FINSEQ_3:25,PARTFUN1:def 6;
   i in dom(fr(#)F) by A4,A6,FINSEQ_3:25;
   hence (fr(#)F).i=fr.Fii*Fii by RLVECT_2:def 7
    .=fv.(Fv/.i)*(Fv/.i) by A1,A2,A8,A9,EUCLID:65
    .=(fv(#)Fv).i by A7,RLVECT_2:def 7;
  end;
  hence thesis by A2,A4,RLVECT_2:def 7;
end;
