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 Th72:
  for T be RealLinearSpace,
      X be set
  holds
    X is Linear_Combination of RLSp2RVSp(T)
      iff
    X is Linear_Combination of T
  proof
    let T be RealLinearSpace,
        X be set;
    set V = RLSp2RVSp(T);

    hereby
      assume X is Linear_Combination of RLSp2RVSp(T);
      then reconsider L = X as Linear_Combination of RLSp2RVSp(T);
      consider S be finite Subset of RLSp2RVSp(T) such that
      A1: for v be Element of RLSp2RVSp(T) st not v in S
          holds L.v = 0. F_Real by VECTSP_6:def 1;
      thus X is Linear_Combination of T by A1,RLVECT_2:def 3;
    end;
    assume X is Linear_Combination of T;
    then reconsider L = X as Linear_Combination of T;
    consider S be finite Subset of T such that
    A2: for v be Element of T st not v in S
        holds L.v = 0 by RLVECT_2:def 3;

    for v be Element of RLSp2RVSp(T) st not v in S
    holds 0. F_Real = L.v by A2;
    hence X is Linear_Combination of RLSp2RVSp(T) by VECTSP_6:def 1;
  end;
