 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 Th24:
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 open
proof
let X be NormedLinearTopSpace;
let x be Point of X;
let r be Real;
let V be Subset of X;
consider RNS be RealNormSpace such that
A1:  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 A1;
 reconsider x1=x as Point of RNS by A1;
assume A2:V = { y where y is Point of X : ||.(x - y).|| < r };
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 A1;
        ||.(x1 - y1).|| < r by Th19,A1,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 A1;
        ||.(x - y).|| < r by Th19,A1,A4;
      hence z in { y where y is Point of X : ||.(x - y).|| < r } by A4;
  end;
then
{ y where y is Point of X : ||.(x - y).|| < r }
= { y where y is Point of RNS : ||.(x1 - y).|| < r } by TARSKI:2;
then V0 is open by NORMSP_2:8,A2;
hence V is open by A1;
end;
