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 st X is open for r be Point of S st r in X
  holds ex g st 0<g & {y where y is Point of S:||.y-r.|| < g} c= X
proof
  let X be Subset of S such that
A1: X is open;
  let r be Point of S;
  assume r in X;
  then consider N be Neighbourhood of r such that
A2: N c= X by A1,Th2;
  consider g such that
A3: 0<g & {y where y is Point of S:||.y-r.|| < g} c= N by NFCONT_1:def 1;
  take g;
  thus thesis by A2,A3;
end;
