
theorem KERX1:
  for X,Y be RealLinearSpace,
      f be Function of X, Y
  st f is homogeneous
  holds f"{0.Y} is non empty
  proof
    let X,Y be RealLinearSpace,
        f be Function of X, Y;
    assume
    A1: f is homogeneous;
    f.(0.X) = f.(0 * 0.X) by RLVECT_1:10
           .= 0 * f.(0.X) by A1,LOPBAN_1:def 5
           .= 0.Y by RLVECT_1:10; then
    f.(0.X) in {0.Y} by TARSKI:def 1;
    hence thesis by FUNCT_2:38;
  end;
