
theorem KLXY1:
  for X,Y be RealLinearSpace,
      f be Function of X, Y
  st f is additive homogeneous
  holds f"{0.Y} is linearly-closed
  proof
    let X,Y be RealLinearSpace,
        f be Function of X, Y;
    assume
    A1: f is additive homogeneous;
    set X1 = f"{0.Y};
    A2: for v,u be Point of X st v in X1 & u in X1 holds v+u in X1
    proof
      let v,u be Point of X;
      assume AS1: v in X1 & u in X1; then
      v in the carrier of X & f.v in {0.Y} by FUNCT_2:38; then
      A3: f.v = 0.Y by TARSKI:def 1;
      A4: u in the carrier of X & f.u in {0.Y} by AS1,FUNCT_2:38;
      f.(v+u) = f.v + f.u by A1
             .= 0.Y + 0.Y by A3,A4,TARSKI:def 1
             .= 0.Y by RLVECT_1:4; then
      f.(v+u) in {0.Y} by TARSKI:def 1;
      hence thesis by FUNCT_2:38;
    end;
    for r be Real, v be Point of X st v in X1 holds r*v in X1
    proof
      let r be Real, v be Point of X;
      assume v in X1; then
      A5: v in the carrier of X & f.v in {0.Y} by FUNCT_2:38;
      f.(r*v) = r * (f.v) by A1,LOPBAN_1:def 5
             .= r * 0.Y by A5,TARSKI:def 1
             .= 0.Y by RLVECT_1:10; then
      f.(r*v) in {0.Y} by TARSKI:def 1;
      hence thesis by FUNCT_2:38;
    end;
    hence thesis by A2;
  end;
