reserve r,s,t,u for Real;

theorem
  for X being LinearTopSpace, x being Point of X, V being a_neighborhood
  of x holds -x+V is a_neighborhood of 0.X
proof
  let X be LinearTopSpace, x be Point of X, V be a_neighborhood of x;
  -x+Int(V) = {-x + v where v is Point of X: v in Int V} & x in Int V by
CONNSP_2:def 1,RUSUB_4:def 8;
  then -x+x in -x+Int(V);
  then
A1: 0.X in -x+Int(V) by RLVECT_1:5;
  -x+Int(V) c= Int(-x+V) by Th37;
  hence thesis by A1,CONNSP_2:def 1;
end;
