 reserve
  S for non empty TopSpace,
  T for LinearTopSpace,
  X for non empty Subset of the carrier of S;
 reserve
    S,T for RealNormSpace,
    X for non empty Subset of the carrier of S;

theorem Th30:
for X being NormedLinearTopSpace,
    V be Subset of X,
    x being Point of X
holds
( V is a_neighborhood of x
  iff
 ( ex r being Real st
  r > 0 & { y where y is Point of X : ||.y-x.|| < r } c= V ))
proof
let X be NormedLinearTopSpace,
    V be Subset of X,
    x be Point of X;
hereby
assume V is a_neighborhood of x;
then consider U being Subset of X such that
A1: U is open and
A2: U c= V and
A3: x in U by CONNSP_2:6;
consider r being Real such that
A4:
 r > 0 & { y where y is Point of X : ||.(x - y).|| < r } c= U
by Th23,A1,A3;
{ y where y is Point of X : ||.(x - y).|| < r }
=
{ y where y is Point of X : ||. y - x.|| < r } by Lm1;
hence ex r being Real st
  r > 0
& { y where y is Point of X : ||.y - x.|| < r } c= V
by A4,A2,XBOOLE_1:1;
end;
given r being Real such that
A5: r > 0
& { y where y is Point of X : ||.(y - x).|| < r } c= V;
reconsider U = { y where y is Point of X : ||.(y - x).|| < r }
 as Subset of X by A5,XBOOLE_1:1;
A6:
{ y where y is Point of X : ||.(y - x).|| < r } =
{ y where y is Point of X : ||.x - y.|| < r } by Lm1;
 ||.(x - x).|| = 0 by NORMSP_1:6; then
x in U by A5;
hence V is a_neighborhood of x by A5,A6,Th24,CONNSP_2:6;
end;
