
theorem Th4:
  for X,x being set, b being Rbag of X st dom b = {x} holds Sum b = b.x
proof
  let X,x be set, b be Rbag of X;
  assume
A1: dom b = {x};
  then
A2: x in dom b by TARSKI:def 1;
  support b c= {x} & {x} c= X by A1,PRE_POLY:37;
  then consider f being FinSequence of REAL such that
A3: f = b*canFS({x}) and
A4: Sum b = Sum f by UPROOTS:14;
   reconsider bx = b.x as Element of REAL by XREAL_0:def 1;
  f = b*<*x*> by A3,FINSEQ_1:94;
  then f = <*bx*> by A2,FINSEQ_2:34;
  hence thesis by A4,FINSOP_1:11;
end;
