reserve a,b,p,r,r1,r2,s,s1,s2,x0,x for Real;
reserve f,g for PartFunc of REAL,REAL;
reserve X,Y for set;

theorem Th3:
  for n being Nat, R being Element of n-tuples_on REAL
  st Sum R=0 & (for i being Element of NAT st i in dom R holds 0 <= R.i) holds
  for i being Element of NAT st i in dom R holds R.i = 0
proof
  let n be Nat, R be Element of n-tuples_on REAL such that
A1: Sum R=0 and
A2: for i being Element of NAT st i in dom R holds 0 <= R.i;
A3: for i being Nat st i in dom R holds 0 <= R.i by A2;
  given k being Element of NAT such that
A4: k in dom R and
A5: R.k <> 0;
  0 <= R.k by A2,A4;
  hence contradiction by A1,A3,A4,A5,RVSUM_1:85;
end;
