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 Th49:
  for V be RealLinearSpace,
      W be strict Subspace of V
  holds
    W is strict Subspace of (Omega). V
  proof
    let V be RealLinearSpace,
        W be strict Subspace of V;
    set V0 = (Omega). V;
    A1: V0
      = RLSStruct(# the carrier of V,
                    the ZeroF of V,
                    the addF of V,
                    the Mult of V #) by RLSUB_1:def 4;

    A2: the carrier of W c= the carrier of V
      & 0. W = 0. V
      & the addF of W = (the addF of V) || (the carrier of W)
      & the Mult of W = (the Mult of V) | [:REAL, the carrier of W:]
        by RLSUB_1:def 2;

    the carrier of V = the carrier of V0
    & 0. V = 0. V0
    & the addF of V = the addF of V0
    & the Mult of V = the Mult of V0 by A1;

    hence thesis by A2,RLSUB_1:def 2;
  end;
