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 Th83:
  for n be non empty Nat
  holds
  RLSp2RVSp(RealVectSpace(Seg n)) = n -VectSp_over F_Real
  proof
    let n be non empty Nat;
    set X = RealVectSpace(Seg n);
    set V = n -VectSp_over F_Real;
    set W = TOP-REAL n;

    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 #)
        = X by EUCLID:def 8;

    A5: 0.(RLSp2RVSp X)
     = 0. (TOP-REAL n) by A3
    .= 0*n by EUCLID:70
    .= 0.(n -VectSp_over F_Real) by A1,A2;

    A6: the addF of (RLSp2RVSp X)
     = the addF of (n -VectSp_over F_Real) by A1,A2,Th51;

    the lmult of (RLSp2RVSp X)
     = n -Mult_over F_Real by Th52
    .= the lmult of (n -VectSp_over F_Real) by PRVECT_1:def 5;
    hence thesis by A3,A5,A6,Lm1;
  end;
