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
  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) iff X is open) by Th2,Th4;
