theorem
  for f being Function of V,W holds f is Homomorphism of V,W iff
  (for x, y being Vector of V holds f.(x+y) = f.x+f.y) &
  for a being Scalar of K, x being Vector of V holds f.(a*x) = a*f.x
proof
  let f be Function of V,W;
  set J = id K;
A1: (for a being Scalar of K, x being Vector of V holds f.(a*x) = a*f.x)
    implies
  for a be Scalar of K, x be Vector of V holds f.(a*x) = J.a*f.x;
  now
    assume
A2: for a being Scalar of K, x being Vector of V holds f.(a*x) = J.a*f .x;
    let a be Scalar of K, x be Vector of V;
    J.a = a;
    hence f.(a*x) = a*f.x by A2;
  end;
  hence thesis by A1,Def17;
end;
