 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 Th32:
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 compact iff W is compact
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 compact;
  for t1 being sequence of RNS st rng t1 c= V0 holds
   ex t2 being sequence of RNS st
      t2 is subsequence of t1
    & t2 is convergent & lim t2 in V0
 proof
   let t1 be sequence of RNS;
    assume A3: rng t1 c= V0;
    reconsider s1=t1 as sequence of X by A1;
   consider s2 being sequence of X such that
   A4:   s2 is subsequence of s1
        & s2 is convergent & lim s2 in V by A2,Th31,A3,A1;
   reconsider t2=s2 as sequence of RNS by A1;
   take t2;
    thus thesis by A4,A1,Th26;
 end;
 hence V0 is compact;
end;
assume
A5: V0 is compact;
  for t1 being sequence of X st rng t1 c= V holds
   ex t2 being sequence of X st
      t2 is subsequence of t1
    & t2 is convergent & lim t2 in V
 proof
   let t1 be sequence of X;
    assume A6: rng t1 c= V;
    reconsider s1=t1 as sequence of RNS by A1;
    consider s2 being sequence of RNS such that
A7: s2 is subsequence of s1
        & s2 is convergent & lim s2 in V0 by A5,A6,A1;
    reconsider t2=s2 as sequence of X by A1;
    take t2;
    t2 is convergent by A7,A1,Th27;
    hence thesis by A7,A1,Th26;
  end;
  hence V is compact by Th31;
end;
