 reserve X for RealUnitarySpace;
 reserve x, y, y1, y2 for Point of X;

theorem Th12:
  for S be RealUnitarySpace,
      X be Subset of S holds
  X is closed Subset of TopSpaceNorm RUSp2RNSp S
   iff
for s1 being sequence of S st rng s1 c= X & s1 is convergent holds
lim s1 in X
proof
  let S be RealUnitarySpace,
      X be Subset of S;
  reconsider Y = X as Subset of TopSpaceNorm RUSp2RNSp S;
  hereby assume X is closed Subset of TopSpaceNorm RUSp2RNSp S;
 then
 reconsider Y = X as closed Subset of TopSpaceNorm RUSp2RNSp S;
A1: for s being sequence of S
st ( for n being Nat holds s . n in Y ) & s is convergent holds
lim s in Y by Th11;
thus
for s1 being sequence of S st rng s1 c= X & s1 is convergent holds
lim s1 in X
proof
  let s1 be sequence of S;
  assume that A2: rng s1 c= X and
   A3: s1 is convergent;
now let n be Nat;
   s1.n in rng s1 by FUNCT_2:4,ORDINAL1:def 12;
  hence s1.n in X by A2;
end;
hence lim s1 in X by A1,A3;
end;
end;
assume
A4: for s1 being sequence of S st rng s1 c= X & s1 is convergent holds
lim s1 in X;
 for s being sequence of S
st ( for n being Nat holds s . n in Y ) & s is convergent holds
lim s in Y
proof
let s be sequence of S;
assume that
 A5: for n being Nat holds s . n in Y
   and
 A6: s is convergent;
 rng s c= X
 proof let y be object;
  assume y in rng s; then
  consider n be object such that
    A7: n in dom s & y=s.n by FUNCT_1:def 3;
    reconsider n as Nat by A7;
    thus y in X by A7,A5;
 end;
hence lim s in Y by A4,A6;
end;
hence thesis by Th11;
end;
