
theorem
  for X being non empty LinearTopSpace,
      x being Point of X,
      O being local_base of X holds
  {x+U where U is Subset of X:U in O & U is a_neighborhood of 0.X} =
  {x+U where U is Subset of X:U in O & U in NeighborhoodSystem 0.X}
  proof
    let X be non empty LinearTopSpace,
    x be Point of X, O be local_base of X;
    now
      let t be object;
      assume t in {x+U where U is Subset of X:U in O &
      U is a_neighborhood of 0.X};
      then consider U1 be Subset of X such that
A1:   t=x+U1 and
A2:   U1 in O and
A3:   U1 is a_neighborhood of 0.X;
      U1 in NeighborhoodSystem 0.X by A3,YELLOW19:2;
      hence t in {x+U where U is Subset of X:U in O &
      U in NeighborhoodSystem 0.X} by A1,A2;
    end;
    then
A4:  {x+U where U is Subset of X:U in O & U is a_neighborhood of 0.X} c=
    {x+U where U is Subset of X:U in O & U in NeighborhoodSystem 0.X};
    now
      let t be object;
      assume t in {x+U where U is Subset of X:U in O &
      U in NeighborhoodSystem 0.X};
      then consider U1 be Subset of X such that
A5:   t=x+U1 and
A6:   U1 in O and
A7:   U1 in NeighborhoodSystem 0.X;
      U1 is a_neighborhood of 0.X by A7,YELLOW19:2;
      hence t in {x+U where U is Subset of X:U in O &
      U is a_neighborhood of 0.X} by A5,A6;
    end;
    then {x+U where U is Subset of X:U in O & U in NeighborhoodSystem 0.X} c=
    {x+U where U is Subset of X:U in O & U is a_neighborhood of 0.X};
    hence thesis by A4;
end;
