 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
for X being NormedLinearTopSpace
for x being Point of X
for r being Real
for V being Subset of X
st V = { y where y is Point of X : ||.(x - y).|| <= r } holds
V is closed
proof
let X be NormedLinearTopSpace;
let x be Point of X;
let r be Real;
let V be Subset of X;
assume
A1: V = { y where y is Point of X : ||.(x - y).|| <= r };
consider RNS be RealNormSpace such that
A2: RNS = the NORMSTR of X
  & the topology of X = the topology of (TopSpaceNorm RNS) by Def7;
 reconsider V0 = V as Subset of (TopSpaceNorm RNS) by A2;
 reconsider x1=x as Point of RNS by A2;
 reconsider V1 = V as Subset of (LinearTopSpaceNorm RNS)
        by A2,NORMSP_2:def 4;
for z be object holds
z in { y where y is Point of X : ||.(x - y).|| <= r }
iff z in { y where y is Point of RNS : ||.(x1 - y).|| <= r }
   proof
    let z be object;
   hereby assume z in { y where y is Point of X
             : ||.(x - y).|| <= r }; then
     consider y be Point of X such that
       A3:y=z & ||.(x - y).|| <= r;
      reconsider y1=y as Point of RNS by A2;
      ||.(x1 - y1).|| <= r by Th19,A2,A3;
      hence z in { y where y is Point of RNS
             : ||.(x1 - y).|| <= r } by A3;
  end;
 assume z in { y where y is Point of RNS : ||.(x1 - y).|| <= r }; then
     consider y1 be Point of RNS such that
       A4:y1=z & ||.(x1 - y1).|| <= r;
      reconsider y=y1 as Point of X by A2;
        ||.(x - y).|| <= r by Th19,A2,A4;
      hence z in { y where y is Point of X : ||.(x - y).|| <= r } by A4;
  end; then
V0 is closed by NORMSP_2:24,A1,TARSKI:2; then
([#] (TopSpaceNorm RNS)) \ V0 is open; then
([#] X) \ V is open by A2;
hence V is closed;
end;
