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 Th20:
  for Lv be Linear_Combination of RealVectSpace(Seg n),
      Lr be Linear_Combination of TOP-REAL n st Lr = Lv
  holds Sum Lr = Sum Lv
proof
  set V=RealVectSpace(Seg n);
  set T=TOP-REAL n;
  let Lv be Linear_Combination of V;
  let Lr be Linear_Combination of T such that
A1: Lr=Lv;
  consider F be FinSequence of the carrier of T such that
A2: (F is one-to-one) & rng F=Carrier(Lr) and
A3: Sum Lr=Sum(Lr(#)F) by RLVECT_2:def 8;
  reconsider F1=F as FinSequence of the carrier of V by Lm1;
A4: Lr(#)F=Lv(#)F1 by A1,Th18;
  thus Sum Lv=Sum(Lv(#)F1) by A1,A2,RLVECT_2:def 8
   .=Sum Lr by A3,A4,Th19;
end;
