reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem Th11:
  for X be set
  for n be Nat
  holds
    X is Linear_Combination of REAL-NS n
      iff
    X is Linear_Combination of TOP-REAL n
  proof
    let X be set;
    let n be Nat;

    A1: the carrier of TOP-REAL n
      = the carrier of REAL-NS n by Th4;

    hereby
      assume X is Linear_Combination of REAL-NS n; then
      reconsider Lr = X as Linear_Combination of REAL-NS n;
      consider T be finite Subset of REAL-NS n such that
      A2: for v be Element of REAL-NS n st not v in T holds
          Lr.v = 0 by RLVECT_2:def 3;
      thus X is Linear_Combination of TOP-REAL n by A1,A2,RLVECT_2:def 3;
    end;

    assume X is Linear_Combination of TOP-REAL n; then
    reconsider Lr = X as Linear_Combination of TOP-REAL n;

    consider T be finite Subset of TOP-REAL n such that
    A3: for v be Element of TOP-REAL n st not v in T holds
        Lr.v = 0 by RLVECT_2:def 3;
    thus X is Linear_Combination of REAL-NS n by A1,A3,RLVECT_2:def 3;
  end;
