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 Th30:
  for X be object holds
    X is Subspace of REAL-NS n
      iff
    X is Subspace of TOP-REAL n
  proof
    let X be object;

    A1: the addF of REAL-NS n
      = the addF of (the RLSStruct of REAL-NS n)
     .= the addF of (the RLSStruct of TOP-REAL n) by Th1
     .= the addF of TOP-REAL n;

    A2: the Mult of REAL-NS n
       = the Mult of (the RLSStruct of REAL-NS n)
      .= the Mult of (the RLSStruct of TOP-REAL n) by Th1
      .= the Mult of TOP-REAL n;

    hereby
      assume X is Subspace of REAL-NS n; then
      reconsider V = X as Subspace of REAL-NS n;

      A3: the carrier of V c= the carrier of REAL-NS n
        & 0. V = 0. (REAL-NS n)
        & the addF of V = (the addF of (REAL-NS n)) || the carrier of V
        & the Mult of V = (the Mult of (REAL-NS n))
            | [:REAL, the carrier of V:] by RLSUB_1:def 2; then
      A4: the carrier of V c= the carrier of TOP-REAL n by Th4;
      0.V = 0. (TOP-REAL n) by A3,Th6;
      hence X is Subspace of TOP-REAL n by A1,A2,A3,A4,RLSUB_1:def 2;
    end;
    assume X is Subspace of TOP-REAL n;
    then reconsider V = X as Subspace of TOP-REAL n;

    A5: the carrier of V c= the carrier of TOP-REAL n
      & 0. V = 0. (TOP-REAL n)
      & the addF of V = (the addF of (TOP-REAL n)) || the carrier of V
      & the Mult of V = (the Mult of (TOP-REAL n))
          | [:REAL, the carrier of V:] by RLSUB_1:def 2;

    A6: the carrier of V c= the carrier of REAL-NS n by A5,Th4;
    0.V = 0. (REAL-NS n) by A5,Th6;
    hence X is Subspace of REAL-NS n by A1,A2,A5,A6,RLSUB_1:def 2;
  end;
