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 n be non empty Nat
  for At be Subset of TOP-REAL n,
      Ar be Subset of n -VectSp_over F_Real
    st At = Ar
  holds
      the carrier of Lin At = the carrier of (Lin Ar)
    & 0.(Lin At) = 0.(Lin Ar)
    & the addF of Lin At = the addF of (Lin Ar)
    & the Mult of Lin At = the lmult of (Lin Ar)
  proof
    let n be non empty Nat;
    let At be Subset of TOP-REAL n,
        Ar be Subset of n -VectSp_over F_Real;

    assume
    A1: At = Ar;
    set V = TOP-REAL n;
    set W = n -VectSp_over F_Real;
    set Lt = Lin At;
    set Lr = Lin Ar;

    A2: the carrier of Lr c= the carrier of W
      & 0. Lr = 0. W
      & the addF of Lr = (the addF of W) || the carrier of Lr
      & the lmult of Lr = (the lmult of W) |
        [: the carrier of F_Real, the carrier of Lr:] by VECTSP_4:def 2;

    A3: the carrier of Lt c= the carrier of V
      & 0. Lt = 0. V
      & the addF of Lt = (the addF of V) || the carrier of Lt
      & the Mult of Lt = (the Mult of V) | [:REAL, the carrier of Lt:]
        by RLSUB_1:def 2;

    thus
    A4: the carrier of Lt
      = [#]Lin At
     .= [#]Lin Ar by A1,MATRTOP2:6
     .= the carrier of Lr;
    A5: the addF of Lt
      = (the addF of W) || the carrier of Lr by A3,A4,Th54
     .= the addF of Lr by VECTSP_4:def 2;
    the Mult of Lt
      = (the lmult of W) | [:REAL, the carrier of Lr:] by A3,A4,Th54
     .= the lmult of Lr by VECTSP_4:def 2;
    hence thesis by A2,A3,A5,Th54;
  end;
