reserve n,m for Nat;
reserve x,X,X1 for set;
reserve s,g,r,p for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve s1,s2 for sequence of S;
reserve x0,x1,x2 for Point of S;
reserve Y for Subset of S;

theorem Th4:
  for N being Neighbourhood of x0 holds x0 in N
proof
  let N be Neighbourhood of x0;
  consider g be Real such that
A1: 0<g and
A2: {z where z is Point of S : ||.z-x0 .|| < g} c= N by Def1;
  ||.x0-x0 .|| = ||.0.S.|| by RLVECT_1:15
    .=0 by NORMSP_1:1;
  then x0 in {z where z is Point of S : ||.z-x0 .|| < g} by A1;
  hence thesis by A2;
end;
