 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,
    RNS be RealNormSpace,
    V being Subset of X,
    W being Subset of RNS st
  RNS = the NORMSTR of X
  & the topology of X = the topology of (TopSpaceNorm RNS)
  & V = W
holds
V is closed iff W is closed
proof
  let X be NormedLinearTopSpace,
      RNS be RealNormSpace,
      V be Subset of X,
      V0 be Subset of RNS;
assume A1:
  RNS = the NORMSTR of X
  & the topology of X = the topology of (TopSpaceNorm RNS)
  & V = V0;
hereby assume
     A2: V is closed;
  for s2 being sequence of RNS st rng s2 c= V0 & s2 is convergent holds
   lim s2 in V0
 proof
   let s2 be sequence of RNS;
    reconsider s1=s2 as sequence of X by A1;
    assume A3: rng s2 c= V0 & s2 is convergent; then
    A4: s1 is convergent by Th27,A1;
    then lim s1 = lim s2 by Th26,A1;
    hence lim s2 in V0 by A1,A2,Th28,A3,A4;
 end;
 hence V0 is closed;
 end;
assume A5:V0 is closed;
  for s2 being sequence of X st rng s2 c= V & s2 is convergent holds
   lim s2 in V
 proof
   let s2 be sequence of X;
    assume A6: rng s2 c= V & s2 is convergent;
    reconsider s1=s2 as sequence of RNS by A1;
    A7: s1 is convergent by Th27,A1,A6;
    lim s1 = lim s2 by Th26,A1,A6;
    hence lim s2 in V by A1,A5,A6,A7;
 end;
 hence V is closed by Th28;
end;
