reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;

theorem Th7:
  j <> 0 & rr = 0 iff Product(j|->rr) = 0
proof
  set f = j|->rr;
A1: dom f = Seg j by FUNCOP_1:13;
  hereby
    assume that
A2: j <> 0 and
A3: rr = 0;
    ex n be Nat st j = n + 1 by A2,NAT_1:6;
    then 1 <= j by NAT_1:11;
    then
A4: 1 in Seg j by FINSEQ_1:1;
    then f.1 = 0 by A3,FUNCOP_1:7;
    hence Product f = 0 by A1,A4,RVSUM_1:103;
  end;
  assume Product f = 0;
  then ex n being Nat st n in dom f & f.n = 0 by RVSUM_1:103;
  hence thesis by A1,FUNCOP_1:7;
end;
