
theorem Th16:
  for V being RealLinearSpace, L being Linear_Combination of V st
  L is circled holds Carrier L <> {}
proof
  let V be RealLinearSpace, L be Linear_Combination of V;
  assume that
A1: L is circled and
A2: Carrier L = {};
  consider F being FinSequence of the carrier of V such that
A3: F is one-to-one & rng F = Carrier L and
A4: ex f being FinSequence of REAL st len f = len F & Sum(f) = 1 & for n
  being Nat st n in dom f holds f.n = L.(F.n) & f.n >= 0 by A1;
  consider f being FinSequence of REAL such that
A5:  len f = len F & Sum(f) = 1 & for n
  being Nat st n in dom f holds f.n = L.(F.n) & f.n >= 0 by A4;
  len F = 0 by A2,A3,CARD_1:27,FINSEQ_4:62;
  then f = <*>the carrier of V by A5;
  hence contradiction by A5,RVSUM_1:72;
end;
