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 Th73:
  for T be RealLinearSpace,
     Lv be Linear_Combination of RLSp2RVSp(T),
     Lr be Linear_Combination of T
    st Lr = Lv
  holds Carrier Lr = Carrier Lv
  proof
    let T be RealLinearSpace,
      Lv be Linear_Combination of RLSp2RVSp(T),
      Lr be Linear_Combination of T;

    assume
    A1: Lr = Lv;

    thus Carrier Lr c= Carrier Lv
    proof
      let x be object;
      assume
      A2: x in Carrier Lr;
      then reconsider v = x as Element of T;
      reconsider u = v as Element of RLSp2RVSp(T);
      Lv.u <> 0. F_Real by A1,A2,RLVECT_2:19;
      hence x in Carrier Lv by VECTSP_6:1;
    end;
    let x be object;
    assume x in Carrier Lv;
    then consider u be Element of RLSp2RVSp(T) such that
    A3: x = u and
    A4: Lv.u <> 0. F_Real by VECTSP_6:1;
    thus x in Carrier Lr by A1,A4,RLVECT_2:19,A3;
  end;
