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 Th54:
  for n be non empty Nat holds
  the carrier of (TOP-REAL n) = the carrier of (n -VectSp_over F_Real)
  & 0. (TOP-REAL n) = 0. (n -VectSp_over F_Real)
  & the addF of (TOP-REAL n) = the addF of (n -VectSp_over F_Real)
  & the Mult of (TOP-REAL n) = the lmult of (n -VectSp_over F_Real)
  proof
    let n be non empty Nat;
    set V = n -VectSp_over F_Real;
    set W = TOP-REAL n;

    thus the carrier of (TOP-REAL n)
    = the carrier of (n -VectSp_over F_Real) by Lm1;

    A1: n -Group_over F_Real
      = addLoopStr(# n -tuples_on the carrier of F_Real,
                     product(the addF of F_Real,n),
                     n |-> (0. F_Real) #) by PRVECT_1:def 3;

    A2: addLoopStr(# the carrier of V,
                     the addF of V,
                     the ZeroF of V #)
        = n -Group_over F_Real
      & the lmult of V = n -Mult_over F_Real
        by PRVECT_1:def 5;

    A3: RLSStruct(# the carrier of W,
                    the ZeroF of W,
                    the addF of W,
                    the Mult of W #)
      = RealVectSpace Seg n by EUCLID:def 8;

    thus 0. (TOP-REAL n)
     = 0* n by EUCLID:70
    .= 0.(n -VectSp_over F_Real) by A1,A2;

    thus the addF of TOP-REAL n
    = the addF of (n -VectSp_over F_Real) by A1,A2,A3,Th51;

    thus the Mult of TOP-REAL n
     = n -Mult_over F_Real by A3,Th52
    .= the lmult of (n -VectSp_over F_Real) by PRVECT_1:def 5;
  end;
