reserve m, n for Nat;

theorem Th12:
  for N being Rbag of NAT st support N = {n} holds Sum N = N.n
proof
  let N be Rbag of NAT;
   reconsider Nn = N.n as Element of REAL by XREAL_0:def 1;
   reconsider F = <*Nn*> as FinSequence of REAL;
  assume
A1: support N = {n};
  {n} c= dom N by PRE_POLY:37,A1;
  then n in dom N by ZFMISC_1:31;
  then
A2: F = N * <*n*> by FINSEQ_2:34
    .= N * canFS(support N) by A1,FINSEQ_1:94;
   Sum F = N.n by FINSOP_1:11;
  hence thesis by A2,UPROOTS:def 3;
end;
