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 T be RealLinearSpace
  for X be set
  for U be Subspace of RLSp2RVSp(T)
  for W be Subspace of T
    st [#] U = [#] W
  holds
    X is Linear_Combination of U
      iff
    X is Linear_Combination of W
  proof
    let T be RealLinearSpace;
    let X be set;
    let U be Subspace of RLSp2RVSp(T);
    let W be Subspace of T;
    assume
    A1: [#] U = [#] W;
    hereby
      assume X is Linear_Combination of U;
      then reconsider L = X as Linear_Combination of U;
      ex S be finite Subset of U
      st for v be Element of U st not v in S
         holds L.v = 0. F_Real by VECTSP_6:def 1;
      hence X is Linear_Combination of W by A1,RLVECT_2:def 3;
    end;
    assume X is Linear_Combination of W;
    then reconsider L = X as Linear_Combination of W;
    consider S be finite Subset of W such that
    A2: for v be Element of W st not v in S
        holds L.v = 0 by RLVECT_2:def 3;

    for v be Element of U st not v in S
    holds 0. F_Real = L.v by A1, A2;
    hence X is Linear_Combination of U by A1, VECTSP_6:def 1;
  end;
