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 Th3:
  for g be Real st 0<g holds
  {y where y is Point of S: ||.y-x0 .|| < g} is Neighbourhood of x0
proof
  let g be Real such that
A1: g >0;
  set N= {y where y is Point of S : ||.y-x0 .|| < g};
  N c= the carrier of S
  proof
    let x be object;
    assume x in {y where y is Point of S : ||.y-x0 .|| < g};
    then ex y be Point of S st x=y & ||.y-x0 .|| < g;
    hence thesis;
  end;
  hence thesis by A1,Def1;
end;
