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

theorem
  for n being Element of NAT holds dim (REAL-US n) = n
proof
  let n be Element of NAT;
  reconsider B=RN_Base n as Subset of REAL-US n by REAL_NS1:def 6;
  for I being Basis of REAL-US n holds n = card I
  proof
    let I be Basis of REAL-US n;
    B is Basis of REAL-US n by Th48;
    then card B=card I by RUSUB_4:5;
    hence n = card I by Lm5;
  end;
  hence thesis by RUSUB_4:def 2;
end;
