
theorem Th3:
  for F be FinSequence of REAL st
  for i be Element of NAT st i in dom F holds F.i = 0 holds Sum F = 0
proof
  let F be FinSequence of REAL;
  set i = len F;
  set R1 = i|-> 0;
  reconsider R2 = F as Element of i-tuples_on REAL by FINSEQ_2:92;
A1: Seg len F = dom F by FINSEQ_1:def 3;
  assume
A2: for i be Element of NAT st i in dom F holds 0 = F.i;
A3: for j be Nat st j in Seg i holds R1.j = R2.j by A2,A1;
  len R1 = i by CARD_1:def 7;
  then dom R1 = dom R2 by FINSEQ_3:29;
  then R1 = R2 by A1,A3,FINSEQ_1:13;
  hence thesis by RVSUM_1:81;
end;
