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

theorem Th34:
  for X being Subspace of RealVectSpace(Seg n), x being Element of
  REAL n, a being Real st x in the carrier of X holds a*x in the carrier of X
proof
  let X be Subspace of RealVectSpace(Seg n), x be Element of REAL n,a be Real;
  reconsider x1=x as Element of RealVectSpace(Seg n) by FINSEQ_2:93;
  assume x in the carrier of X;
  then
A1: x in X;
  reconsider aa=a as Element of REAL by XREAL_0:def 1;
  aa*x=aa*x1;
  then a*x in X by A1,RLSUB_1:21;
  hence a*x in the carrier of X;
end;
