reserve x,y for set,
  n,m for Nat,
  r,s for Real;

theorem
  for R be real-valued FinSequence holds (0*R)"{0} = dom R
proof
  let R be real-valued FinSequence;
  reconsider R1 = R as FinSequence of REAL by RVSUM_1:145;
A1: Seg len(0*R) = dom(0*R) by FINSEQ_1:def 3;
A2: len(0*R1) = len R1 & dom R1 = Seg len R1 by FINSEQ_1:def 3,FINSEQ_2:33;
  hence (0*R)"{0} c= dom R by A1,RELAT_1:132;
  let x be object;
  assume
A3: x in dom R;
  then reconsider i = x as Element of NAT;
  (0*R).i = 0*R.i by RVSUM_1:44
    .= 0;
  then (0*R).i in {0} by TARSKI:def 1;
  hence thesis by A2,A1,A3,FUNCT_1:def 7;
end;
