reserve T1,T2,T3 for TopSpace,
  A1 for Subset of T1, A2 for Subset of T2, A3 for Subset of T3;
reserve n,k for Nat;
reserve M,N for non empty TopSpace;
reserve p,q,p1,p2 for Point of TOP-REAL n;
reserve r for Real;

theorem
  RN_Base n is Basis of TOP-REAL n
proof
  set A = RN_Base n;
  set T = TOP-REAL n;
  RN_Base n c= REAL n;
  then A1: RN_Base n c= the carrier of T by EUCLID:22;
  reconsider A as finite Subset of T by EUCLID:22;
  reconsider B = RN_Base n as Subset of RealVectSpace(Seg n) by A1,Lm1;
  B is Basis of RealVectSpace(Seg n) by EUCLID_7:45;
  then B is linearly-independent by RLVECT_3:def 3;
  then A2: A is linearly-independent by Th21;
  reconsider V1 = (Omega).T as RealLinearSpace;
  for v being VECTOR of T st v in Lin A holds v in V1
  proof
    let v be VECTOR of T;
    assume v in Lin A;
    v in the RLSStruct of T;
    hence v in V1 by RLSUB_1:def 4;
  end;
  then reconsider X = Lin A as Subspace of V1 by RLSUB_1:29;
  for x being object holds x in the carrier of X iff x in the carrier of V1
  proof
    let x be object;
    hereby
      assume x in the carrier of X;
      then x in X;
      then x in V1 by RLSUB_1:9;
      hence x in the carrier of V1;
    end;
    assume x in the carrier of V1;
    then x in the RLSStruct of T by RLSUB_1:def 4;
    then reconsider x0 = x as Element of TOP-REAL n;
    ex l being Linear_Combination of A st x0 = Sum l by Th22;
    then x in the set of all Sum l where l is Linear_Combination of A;
    hence x in the carrier of X by RLVECT_3:def 2;
  end;
  then Lin A = (Omega).T by TARSKI:2,EUCLID_7:9;
  then Lin A = the RLSStruct of T by RLSUB_1:def 4;
  hence RN_Base n is Basis of T by A2,RLVECT_3:def 3;
end;
