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
  for X be set,
      U be Subspace of REAL-NS n,
      W be Subspace of n -VectSp_over F_Real
    st [#] U = [#] W
  holds
    X is Linear_Combination of U
      iff
    X is Linear_Combination of W
  proof
    let X be set;
    let U be Subspace of REAL-NS n;
    let W be Subspace of n -VectSp_over F_Real;
    assume
    A1: [#] U = [#] W;
    reconsider S=U as Subspace of TOP-REAL n by Th30;
    X is Linear_Combination of S
          iff
        X is Linear_Combination of W by A1,MATRTOP2:11;
    hence thesis;
  end;
