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

theorem Th44:
  for B being Subset of RealVectSpace(Seg n) st B=RN_Base n holds
  B is Basis of RealVectSpace(Seg n)
proof
  let B be Subset of RealVectSpace(Seg n);
  set V= RealVectSpace(Seg n);
  assume
A1: B=RN_Base n;
  then reconsider B1 = B as R-orthonormal Subset of V;
A2: the carrier of Lin B = the set of all
 (Sum l) where l is Linear_Combination of B  by RLVECT_3:def 2;
A3: now
    assume not the carrier of V c= the carrier of Lin B;
    then consider x being object such that
A4: x in the carrier of V and
A5: not x in the carrier of Lin B;
    reconsider x0=x as Element of V by A4;
    ex l being Linear_Combination of B st x0=Sum l by A1,Th42;
    hence contradiction by A2,A5;
  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 A2;
    hence x in the carrier of V;
  end;
  then the carrier of Lin B=the carrier of V by A3,XBOOLE_0:def 10;
  then Lin B = V by Th8;
  then B1 is Basis of RealVectSpace(Seg n) by RLVECT_3:def 3;
  hence thesis;
end;
