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 Th84:
  for V,W be RealLinearSpace,
      X be set
  holds
    X is LinearOperator of V,W
      iff
    X is linear-transformation of RLSp2RVSp(V),RLSp2RVSp(W)
  proof
    let V,W be RealLinearSpace,
        X be set;
    hereby
      assume X is LinearOperator of V,W; then
      reconsider T = X as LinearOperator of V,W;
      reconsider f = T as Function of RLSp2RVSp(V),RLSp2RVSp(W);

      for x, y be Element of RLSp2RVSp(V)
      holds f . (x + y) = (f . x) + (f . y)
      proof
        let x, y be Element of RLSp2RVSp (V);

        A1: T is additive;
        reconsider x0 = x, y0 = y as Element of V;
        thus f . (x + y) = T.(x0 + y0)
        .= (T.x0) + (T.y0) by A1
        .= f.x + f.y;
      end;
      then
      A2: f is additive;

      for a be Scalar of F_Real
      for x be Vector of RLSp2RVSp(V)
      holds f . (a * x) = a * (f . x)
      proof
        let a be Scalar of F_Real;
        let x be Element of RLSp2RVSp (V);
        reconsider x0 = x as Element of V;
        reconsider a0 = a as Element of REAL;
        thus f.(a * x)
         = T.(a0 * x0)
        .= a0 * T.x0 by LOPBAN_1:def 5
        .= a * f.x;
      end;
      hence X is linear-transformation of RLSp2RVSp(V),RLSp2RVSp(W)
        by MOD_2:def 2,A2;
    end;
    assume X is linear-transformation of RLSp2RVSp(V),RLSp2RVSp(W); then
    reconsider T = X as linear-transformation of RLSp2RVSp(V),RLSp2RVSp(W);
    reconsider f = T as Function of V,W;
    for x, y be Element of V
    holds f . (x + y) = (f . x) + (f . y)
    proof
      let x, y be Element of V;

      A3: T is additive;
      reconsider x0 = x, y0 = y as Element of RLSp2RVSp(V);
      thus f . (x + y)
       = T.(x0 + y0)
      .= (T.x0) + (T.y0) by A3
      .= f.x + f.y;
    end;
    then
    A4: f is additive;
    for a be Real,
        x be VECTOR of V
    holds f . (a * x) = a * (f . x)
    proof
      let a be Real;
      let x be VECTOR of V;
      reconsider x0 = x as Element of RLSp2RVSp (V);
      reconsider a0 = a as Element of F_Real by XREAL_0:def 1;
      thus f.(a * x)
       = T.(a0 * x0)
      .= a0 * T.x0 by MOD_2:def 2
      .= a * f.x;
    end;
    then f is homogeneous;
    hence X is LinearOperator of V,W by A4;
  end;
