
theorem KLXY1:
  for X be RealUnitarySpace,
      f be Function of X,REAL
  st f is additive homogeneous
  holds f"{0} is linearly-closed
proof
  let X be RealUnitarySpace,
      f be Function of X,REAL;
  assume
A1: f is additive homogeneous;
    set X1 = f"{0};
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} by FUNCT_2:38; then
A3: f.v = 0 by TARSKI:def 1;
A4: u in the carrier of X & f.u in {0} by AS1,FUNCT_2:38;
    f.(v+u) = f.v + f.u by A1,HAHNBAN:def 2
           .= 0 + 0 by A3,A4,TARSKI:def 1; then
    f.(v+u) in {0} 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} by FUNCT_2:38;
    f.(r*v) = r * (f.v) by A1,HAHNBAN:def 3
           .= r * 0 by A5,TARSKI:def 1; then
    f.(r*v) in {0} by TARSKI:def 1;
    hence thesis by FUNCT_2:38;
  end;
  hence thesis by A2;
end;
