theorem Th3:
  for g be Real st 0<g holds
  {y where y is Point of S: ||.y-x0 .|| < g} is Neighbourhood of x0
proof
  let g be Real such that
A1: g >0;
  set N= {y where y is Point of S : ||.y-x0 .|| < g};
  N c= the carrier of S
  proof
    let x be object;
    assume x in {y where y is Point of S : ||.y-x0 .|| < g};
    then ex y be Point of S st x=y & ||.y-x0 .|| < g;
    hence thesis;
  end;
  hence thesis by A1,Def1;
end;
