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

theorem Th48:
  for n being Element of NAT holds RN_Base n is Basis of REAL-US n
proof
  let n be Element of NAT;
  reconsider B = RN_Base n as Subset of REAL-US n by REAL_NS1:def 6;
  set V = REAL-US n;
A1: the carrier of Lin B = the set of all
Sum l where l is Linear_Combination of B  by RUSUB_3:def 1;
A2: now
    assume not the carrier of V c= the carrier of Lin B;
    then consider x being object such that
A3: x in the carrier of V and
A4: not x in the carrier of Lin B;
    reconsider x0=x as Element of V by A3;
    ex l being Linear_Combination of B st x0=Sum l by Th43;
    hence contradiction by A1,A4;
  end;
  the carrier of Lin B c= the carrier of V
  proof
    let x be object;
    assume x in the carrier of Lin B;
    then ex l being Linear_Combination of B st x = Sum l by A1;
    hence x in the carrier of V;
  end;
  then the carrier of Lin B=the carrier of V by A2,XBOOLE_0:def 10;
  then Lin B = V by RUSUB_1:26;
  hence thesis by RUSUB_3:def 2;
end;
