reserve n,k for Element of NAT;
reserve x,y,X for set;
reserve g,r,p for Real;
reserve S for RealNormSpace;
reserve rseq for Real_Sequence;
reserve seq,seq1 for sequence of S;
reserve x0 for Point of S;
reserve Y for Subset of S;

theorem Th4:
  for X being Subset of S holds ((for r be Point of S st r in X
  holds ex N be Neighbourhood of r st N c= X) implies X is open )
proof
  let X be Subset of S;
  assume that
A1: for r be Point of S st r in X holds ex N be Neighbourhood of r st N
  c= X and
A2: not X is open;
  not X` is closed by A2;
  then consider s1 be sequence of S such that
A3: rng s1 c= X` and
A4: s1 is convergent and
A5: not lim s1 in X`;
  consider N be Neighbourhood of (lim s1) such that
A6: N c= X by A1,A5,SUBSET_1:29;
  consider g be Real such that
A7: 0<g and
A8: {y where y is Point of S:||.y-(lim s1).|| < g} c= N by NFCONT_1:def 1;
  consider n being Nat such that
A9: for m be Nat st n<=m holds ||.(s1.m) - (lim s1).||<g by A4,A7,
NORMSP_1:def 7;
  n in NAT by ORDINAL1:def 12;
  then n in dom s1 by FUNCT_2:def 1;
  then
A10: s1.n in rng s1 by FUNCT_1:def 3;
  ||.s1.n - (lim s1).|| <g by A9;
  then s1.n in {y where y is Point of S:||.y-(lim s1).|| < g};
  then s1.n in N by A8;
  hence contradiction by A3,A6,A10,XBOOLE_0:def 5;
end;
