reserve r,s,t,u for Real;

theorem Th32:
  for X being LinearTopSpace, a being Real, x being Point of X, V
being a_neighborhood of a*x
 ex r being positive Real, W being a_neighborhood of
  x st for s being Real st |.s-a.| < r holds s*W c= V
proof
  let X be LinearTopSpace, a be Real, x be Point of X, V be a_neighborhood of
  a*x;
  a*x in Int(V) by CONNSP_2:def 1;
  then consider r being positive Real, W being Subset of X such that
A1: W is open and
A2: x in W and
A3: for s being Real st |.s-a.| < r holds s*W c= Int(V) by Def9;
  Int(W) = W by A1,TOPS_1:23;
  then reconsider W as a_neighborhood of x by A2,CONNSP_2:def 1;
  take r;
  take W;
  let s be Real;
  assume |.s-a.| < r;
  then
A4: s*W c= Int(V) by A3;
  Int(V) c= V by TOPS_1:16;
  hence thesis by A4;
end;
