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 Th1:
  the RLSStruct of TOP-REAL n = the RLSStruct of REAL-NS n
  proof
    set V = the RLSStruct of TOP-REAL n;
    set W = the RLSStruct of REAL-NS n;
    A1: the carrier of W
      = REAL n by REAL_NS1:def 4
     .= the carrier of V by EUCLID:22;
    A2: RLSStruct(# the carrier of (TOP-REAL n),
                    the ZeroF of (TOP-REAL n),
                    the addF of (TOP-REAL n),
                    the Mult of (TOP-REAL n) #)
      = RealVectSpace Seg n by EUCLID:def 8;

    A3: the ZeroF of W
      = 0.(REAL-NS n)
     .= 0* n by REAL_NS1:def 4
     .= 0.(RealVectSpace Seg n) by FINSEQ_2:def 2
     .= the ZeroF of V by EUCLID:def 8;

    A4: Funcs(Seg n,REAL) = REAL n by FINSEQ_2:93;

    for x,y be Element of REAL n
    holds (Euclid_add n).(x,y) = (RealFuncAdd Seg n).(x,y)
    proof
      let x,y be Element of REAL n;
      reconsider x1 = x, y1 = y as Point of TOP-REAL n by A1,REAL_NS1:def 4;
      thus (Euclid_add n).(x,y) = x1 + y1 by REAL_NS1:def 1
      .= (RealFuncAdd Seg n).(x,y) by A2;
    end; then
    Euclid_add n = RealFuncAdd Seg n by A4,BINOP_1:2; then
    A5: the addF of W
      = the addF of RealVectSpace Seg n by REAL_NS1:def 4
     .= the addF of V by EUCLID:def 8;

    for x be Element of REAL, y be Element of REAL n holds
      (Euclid_mult n).(x,y) = (RealFuncExtMult Seg n).(x,y)
    proof
    let x be Element of REAL,
        y be Element of REAL n;

    reconsider y1 = y as Point of TOP-REAL n by A1,REAL_NS1:def 4;
    reconsider x1 = x as Real;
    reconsider y2 = y as VECTOR of REAL-NS n by REAL_NS1:def 4;

    thus (Euclid_mult n). (x,y)
     = x1 * y2 by REAL_NS1:def 4
    .= x1 * y1 by REAL_NS1:3
    .= (RealFuncExtMult Seg n).(x,y) by A2;
    end; then
    Euclid_mult n = RealFuncExtMult Seg n by A4,BINOP_1:2; then
    the Mult of W
     = the Mult of RealVectSpace Seg n by REAL_NS1:def 4
    .= the Mult of V by EUCLID:def 8;
    hence thesis by A1,A3,A5;
  end;
