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

theorem Th35:
  for X being Subspace of RealVectSpace(Seg n), x,y being Element of REAL n st
      x in the carrier of X & y in the carrier of X holds
      x+y in the carrier of X
proof
  let X be Subspace of RealVectSpace(Seg n), x,y be Element of REAL n;
  assume that
A1: x in the carrier of X and
A2: y in the carrier of X;
A3: y in X by A2;
  reconsider x1=x,y1=y as Element of RealVectSpace(Seg n) by FINSEQ_2:93;
A4: x1+y1=x+y;
  x in X by A1;
  then x+y in X by A3,A4,RLSUB_1:20;
  hence x+y in the carrier of X;
end;
