 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 Th31:
for X being NormedLinearTopSpace,
    V being Subset of X holds
V is compact iff
for s1 being sequence of X st rng s1 c= V holds
ex s2 being sequence of X st
   s2 is subsequence of s1
 & s2 is convergent & lim s2 in V
proof
let X be NormedLinearTopSpace;
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 V1 = V as Subset of RNS by A1;
hereby assume A2:V is compact;
       for F being Subset-Family of (TopSpaceNorm RNS)
    st F is Cover of V0 & F is open holds
       ex G being Subset-Family of (TopSpaceNorm RNS) st
     ( G c= F & G is Cover of V0 & G is finite )
    proof
      let F0 being Subset-Family of (TopSpaceNorm RNS);
        assume A3: F0 is Cover of V0 & F0 is open;
        reconsider F=F0 as Subset-Family of X by A1;
         A4:
         for P being Subset of X st P in F holds
         P is open
             proof
              let P being Subset of X;
               assume A5:P in F;
         reconsider P0=P as Subset of (TopSpaceNorm RNS) by A1;
               P0 is open by A3,TOPS_2:def 1,A5;
               hence P is open by A1;
             end;
        consider G being Subset-Family of X such that
         A6: G c= F & G is Cover of V & G is finite
           by A2,COMPTS_1:def 4,A3,A4,TOPS_2:def 1;
        reconsider G0=G as Subset-Family of (TopSpaceNorm RNS) by A1;
        take G0;
        thus G0 c= F0 by A6;
        thus G0 is Cover of V0 by A6;
        thus G0 is finite by A6;
     end; then
      V0 is compact by COMPTS_1:def 4; then
   A7:  V1 is compact by TOPMETR4:19;
   thus for s1 being sequence of X st rng s1 c= V holds
    ex s2 being sequence of X st
    s2 is subsequence of s1 & s2 is convergent & lim s2 in V
   proof
      let s1 be sequence of X;
       reconsider t1 =s1 as sequence of RNS by A1;
       assume rng s1 c= V; then
       consider t2 being sequence of RNS such that
         A9: t2 is subsequence of t1
           & t2 is convergent & lim t2 in V1 by A7;
      reconsider s2=t2 as sequence of X by A1;
      take s2;
      thus s2 is subsequence of s1 by A9;
      s2 is convergent by A9,Th27,A1;
      hence s2 is convergent & lim s2 in V by A9,A1,Th26;
   end;
end;
 assume
 A11:for s1 being sequence of X st rng s1 c= V holds
    ex s2 being sequence of X st
    s2 is subsequence of s1 & s2 is convergent & lim s2 in V;
    for s1 being sequence of RNS st rng s1 c= V1 holds
    ex s2 being sequence of RNS st
    s2 is subsequence of s1 & s2 is convergent & lim s2 in V1
   proof
      let s1 be sequence of RNS;
       assume A12: rng s1 c= V1;
       reconsider t1 =s1 as sequence of X by A1;
       consider t2 being sequence of X such that
         A13: t2 is subsequence of t1
           & t2 is convergent & lim t2 in V by A11,A12;
      reconsider s2=t2 as sequence of RNS by A1;
      take s2;
      thus s2 is subsequence of s1 by A13;
     thus
     s2 is convergent & lim s2 in V1 by A13,Th26,A1;
   end; then
   V1 is compact; then
   A14:V0 is compact by TOPMETR4:19;
     for F being Subset-Family of X
    st F is Cover of V & F is open holds
       ex G being Subset-Family of X st
     G c= F & G is Cover of V & G is finite
    proof
      let F be Subset-Family of X;
        assume A15: F is Cover of V & F is open;
        reconsider F0=F as Subset-Family of (TopSpaceNorm RNS) by A1;
A16:    for P0 being Subset of (TopSpaceNorm RNS) st P0 in F0 holds
          P0 is open
        proof
          let P0 be Subset of TopSpaceNorm RNS;
          assume A17:P0 in F0;
          reconsider P=P0 as Subset of X by A1;
          P is open by A15,TOPS_2:def 1,A17;
          hence P0 is open by A1;
        end;
        consider G0 being Subset-Family of (TopSpaceNorm RNS) such that
A18:    G0 c= F0 & G0 is Cover of V0 & G0 is finite
           by A14,COMPTS_1:def 4,A15, A16,TOPS_2:def 1;
        reconsider G=G0 as Subset-Family of X by A1;
        take G;
        thus G c= F by A18;
        thus G is Cover of V by A18;
        thus G is finite by A18;
     end;
     hence V is compact by COMPTS_1:def 4;
end;
