
theorem Th27:
  for X be finite-dimensional RealLinearSpace
  holds
    product <*X*> is finite-dimensional
  & dim(product <*X*>) = dim X
  proof
    let X be finite-dimensional RealLinearSpace;

    consider I be Function of X,(product <*X*>) such that
    A1: I is one-to-one & I is onto and
        for x be Point of X holds I.x = <*x*> and
    A2: for v, w be Point of X holds I.(v + w) = I.v + I.w and
    A3: for v be Point of X for r be Element of REAL
        holds I.(r * v) = r * I.v and
        I.(0.X) = 0.(product <*X*>) by PRVECT_3:11;

    A4: I is additive by A2;

    for x be VECTOR of X for r be Real
    holds I.(r * x) = r * I.x
    proof
      let x be VECTOR of X, r be Real;
      reconsider r0 = r as Element of REAL by XREAL_0:def 1;
      thus I.(r * x)
       = r0 * I.x by A3
      .= r * I.x;
    end;

    then I is LinearOperator of X,product <*X*> by A4,LOPBAN_1:def 5;
    hence thesis by A1,REAL_NS2:88;
  end;
