reserve x,y for set;
reserve a,b for Real;
reserve i,j for Integer;
reserve V for RealLinearSpace;
reserve W1,W2,W3 for Subspace of V;
reserve v,v1,v2,v3,u,w,w1,w2,w3 for VECTOR of V;
reserve A,B,C for Subset of V;
reserve L,L1,L2 for Linear_Combination of V;
reserve l,l1,l2 for Linear_Combination of A;

theorem Th28:
  for RS be RealLinearSpace,
      f be FinSequence of RS st f = (Seg len f) --> 0.RS
    holds Sum(f)=0.RS
proof
  let RS be RealLinearSpace, f be FinSequence of RS;
  assume A1:f = (Seg len f) --> 0.RS;
A2:now let k be Nat, v be Element of RS;
     assume A3: k in dom f & v = f.k; then
     k in Seg (len f) by FINSEQ_1:def 3; then
     f.k = 0.RS by A1,FUNCOP_1:7;
     hence f.k = -v by A3,RLVECT_1:12;
   end;
   len f = len f;
   then Sum(f) = -Sum(f) by A2,RLVECT_1:40;
   hence thesis by RLVECT_1:19;
end;
