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 At be Subset of TOP-REAL n,
      Ar be Subset of REAL-NS n
    st At = Ar
  holds Lin At = Lin Ar
  proof
    let At be Subset of TOP-REAL n,
        Ar be Subset of REAL-NS n;
    assume
    A1: At = Ar;
    set V = TOP-REAL n;
    set W = REAL-NS n;
    set Lt = Lin At;
    set Lr = Lin Ar;

    A2: 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;

    A3: the carrier of Lt
      = [#]Lin At
     .= [#]Lin Ar by A1,Th26
     .= the carrier of Lr;

    A4: the RLSStruct of TOP-REAL n =the RLSStruct of REAL-NS n by Th1;

    then
    the carrier of V = the carrier of W
    & 0.V = 0.W
    & the addF of V = the addF of W
    & the Mult of V = the Mult of W;

    then
    A5: 0.Lt = 0.Lr by A2,RLSUB_1:def 2;
    A6: the addF of Lt
     = (the addF of W) || the carrier of Lr by A3,A4,RLSUB_1:def 2
    .= the addF of Lr by RLSUB_1:def 2;
    the Mult of Lt
     = (the Mult of W) | [:REAL, the carrier of Lr:] by A3,A4,RLSUB_1:def 2
    .= the Mult of Lr by RLSUB_1:def 2;
    hence thesis by A3,A5,A6;
  end;
