reserve i, j, m, n for Nat,
  z, B0 for set,
  f, x0 for real-valued FinSequence;

theorem
  for X being Subspace of RealVectSpace(Seg n), x,y being Element of
REAL n,a,b being Real st x in the carrier of X & y in the carrier of X holds a*
  x+b*y in the carrier of X
proof
  let X be Subspace of RealVectSpace(Seg n), x,y be Element of REAL n,a,b
  being Real;
  assume that
A1: x in the carrier of X and
A2: y in the carrier of X;
A3: b*y in the carrier of X by A2,Th34;
  a*x in the carrier of X by A1,Th34;
  hence a*x+b*y in the carrier of X by A3,Th35;
end;
