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 Th4:
  for F be FinSequence of TOP-REAL n,
      Fv be FinSequence of n-VectSp_over F_Real st Fv = F
  holds Sum F = Sum Fv
proof
  set T=TOP-REAL n;
  set V=n-VectSp_over F_Real;
  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;
   reconsider Fvi1=Fv/.i1,fvi=fv.i as Element of n-tuples_on
     the carrier of F_Real by MATRIX13:102;
   A10: @@Fvi1=Fvi1 & @@fvi=fvi;
   reconsider Fi1=F/.i1 as Element of T;
   assume A11: i1<=len F;
   then A12: i<len F by NAT_1:13;
   1<=i1 by NAT_1:11;
   then A13: i1 in dom F by A11,FINSEQ_3:25;
   then F.i1=F/.i1 by PARTFUN1:def 6;
   then A14: f.i1=f.i+Fi1 by A4,A12;
   A15: Fv/.i1=Fv.i1 by A1,A13,PARTFUN1:def 6;
   then Fvi1=F/.i1 by A1,A13,PARTFUN1:def 6;
   hence f.i1=@fvi+@Fvi1 by A9,A11,A14,EUCLID:64,NAT_1:13
    .=fvi+Fvi1 by A10,MATRTOP1:1
    .=fv.i+Fv/.i1 by MATRIX13:102
    .=fv.i1 by A1,A7,A12,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;
