
theorem IMX2:
  for X,Y be RealLinearSpace,
      f be Function of X, Y
  st f is additive homogeneous
  holds rng f is linearly-closed
  proof
    let X,Y be RealLinearSpace,
        f be Function of X, Y;
    assume
    A1: f is additive homogeneous;
    set Y1 = rng f;
    A2: for v,u be Point of Y st v in Y1 & u in Y1 holds v+u in Y1
    proof
      let v,u be Point of Y;
      assume
      A3: v in Y1 & u in Y1; then
      consider x1 be Element of the carrier of X such that
      A4: v = f.x1 by FUNCT_2:113;
      consider x2 be Element of the carrier of X such that
      A5: u = f.x2 by A3,FUNCT_2:113;
      v+u = f.(x1+x2) by A1,A4,A5;
      hence thesis by FUNCT_2:4;
    end;
    for r be Real, v be Point of Y st v in Y1 holds r*v in Y1
    proof
      let r be Real, v be Point of Y;
      assume v in Y1; then
      consider x be Element of the carrier of X such that
      A6: v = f.x by FUNCT_2:113;
      r*v = f.(r*x) by A1,A6,LOPBAN_1:def 5;
      hence thesis by FUNCT_2:4;
    end;
    hence thesis by A2;
  end;
