
theorem Th21:
  for V being RealLinearSpace, L being Linear_Combination of V st
  L is convex holds Carrier(L) <> {}
proof
  let V be RealLinearSpace;
  let L be Linear_Combination of V;
  assume L is convex;
  then consider F being FinSequence of the carrier of V such that
A1: F is one-to-one & rng F = Carrier L and
A2: 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;
 consider f being FinSequence of REAL such that
A3:  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 A2;
  assume Carrier(L) = {};
  then len F = 0 by A1,CARD_1:27,FINSEQ_4:62;
  then f = <*>REAL by A3;
  hence contradiction by A3,RVSUM_1:72;
end;
