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 Th19:
  for F be FinSequence of TOP-REAL n,
      Fv be FinSequence of RealVectSpace(Seg n) st Fv = F
  holds Sum F = Sum Fv
proof
  set T=TOP-REAL n;
  set V=RealVectSpace(Seg n);
  let F be FinSequence of T;
  let Fv be FinSequence of V such that
A1: Fv=F;
  reconsider T=TOP-REAL n as RealLinearSpace;
  consider f be sequence of the carrier of T such that
A2: Sum F=f.(len F) and
A3: f.0=0.T and
A4: for j be Nat,v be Element of T st j<len F & v=F.(j+1)
     holds f.(j+1)=f.j+v by RLVECT_1:def 12;
  consider fv be sequence of the carrier of V such that
A5: Sum Fv=fv.(len Fv) and
A6: fv.0=0.V and
A7: for j be Nat,v be Element of V st j<len Fv & v=Fv.(j+1)
    holds fv.(j+1)=fv.j+v by RLVECT_1:def 12;
  defpred P[Nat] means $1<=len F implies f.$1=fv.$1;
A8: for i be Nat st P[i] holds P[i+1]
  proof
    let i be Nat such that
A9: P[i];
    set i1=i+1;
A10: 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 Fvi1=Fv/.i1,fvi=fv.i as Element of n-tuples_on
     the carrier of F_Real by A10,Lm1;
   reconsider Fi1=F/.i1 as Element of T;
   assume
A11: i1<=len F;
A13: i1 in dom F by A11,NAT_1:11,FINSEQ_3:25;
   then F.i1=F/.i1 by PARTFUN1:def 6;
   then A14: f.i1=f.i+Fi1 by A4,A11,NAT_1:13;
A15: Fv/.i1=Fv.i1 by A1,A13,PARTFUN1:def 6;
   then Fvi1 = F/.i1 by A1,A13,PARTFUN1:def 6;
   hence f.i1 = fv.i+Fv/.i1 by A9,A11,A14,EUCLID:64,NAT_1:13
    .=fv.i1 by A1,A7,A11, NAT_1:13,A15;
  end;
A16: P[0] by A3,A6,Lm2;
  for n be Nat holds P[n] from NAT_1:sch 2(A16,A8);
  hence thesis by A1,A2,A5;
end;
