reserve r,s,t,u for Real;

theorem Th54:
  for X being LinearTopSpace, x being Point of X, V being
  a_neighborhood of x, r be non zero Real
   holds r*V is a_neighborhood of r*x
proof
  let X be LinearTopSpace, x be Point of X, V be a_neighborhood of x, r be non
  zero Real;
  x in Int(V) by CONNSP_2:def 1;
  then r*x in r*Int(V);
  hence r*x in Int(r*V) by Th51;
end;
