 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 Th23:
for X being NormedLinearTopSpace,
    V being Subset of X holds
 V is open iff for x being Point of X st x in V holds
                 ex r being Real st
 r > 0 & { y where y is Point of X : ||.(x - y).|| < r } c= V
proof
let X be NormedLinearTopSpace,
    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 V1 = V as Subset of TopSpaceNorm RNS by A1;
hereby
 assume V is open; then
  A2:V1 is open by A1;
 let x be Point of X;
    reconsider x1=x as Point of RNS by A1;
    assume x in V; then
    consider r be Real such that
    A3: r > 0 & { y where y is Point of RNS
             : ||.(x1 - y).|| < r } c= V1 by NORMSP_2:7,A2;
    take r;
    thus r > 0 by A3;
    thus { y where y is Point of X : ||.(x - y).|| < r } c= V
   proof
    let z be object;
      assume z in { y where y is Point of X
             : ||.(x - y).|| < r }; then
     consider y be Point of X such that
       A4:y=z & ||.(x - y).|| < r;
      reconsider y1=y as Point of RNS by A1;
        ||.(x1 - y1).|| < r by Th19,A1,A4; then
      y1 in { y where y is Point of RNS : ||.(x1 - y).|| < r };
      hence z in V by A4,A3;
  end;
 end;
assume A5:for x being Point of X st x in V holds
                 ex r being Real st
  r > 0 & { y where y is Point of X : ||.(x - y).|| < r } c= V;
now
 let x be Point of RNS;
    reconsider x1=x as Point of X by A1;
    assume x in V1; then
    consider r be Real such that
    A6: r > 0 & { y where y is Point of X
             : ||.(x1 - y).|| < r } c= V by A5;
    take r;
    thus r > 0 by A6;
    { y where y is Point of RNS : ||.(x - y).|| < r } c= V1
   proof
    let z be object;
      assume z in { y where y is Point of RNS : ||.(x - y).|| < r }; then
     consider y be Point of RNS such that
       A7:y=z & ||.(x - y).|| < r;
      reconsider y1=y as Point of X by A1;
        ||.(x1 - y1).|| < r by Th19,A1,A7;
      then y1 in { y where y is Point of X : ||.(x1 - y).|| < r };
      hence z in V1 by A7,A6;
  end;
hence { y where y is Point of RNS : ||.(x - y).|| < r } c= V1;
end;
then V1 is open by NORMSP_2:7;
hence V is open by A1;
end;
