reserve X for RealNormSpace;

theorem Th7:
  for X be RealNormSpace, V be Subset of TopSpaceNorm X holds V is
  open iff for x be Point of X st x in V
 ex r be Real st r>0 & {y where y
  is Point of X: ||.x-y.|| < r} c= V
proof
  let X be RealNormSpace, V be Subset of TopSpaceNorm X;
A1: now
    assume
A2: for x be Point of X st x in V holds
   ex r be Real st r>0 & {y where
    y is Point of X: ||.x-y.|| < r} c= V;
    now
      let z be Element of MetricSpaceNorm X;
      reconsider x = z as Point of X;
      assume z in V;
      then consider r be Real such that
A3:   r > 0 & {y where y is Point of X: ||.x-y.|| < r} c= V by A2;
      take r;
      ex t be Point of X st t = z & Ball(z,r) = {y where y is Point of X:
      ||.t-y.|| < r} by Th2;
      hence r > 0 & Ball(z,r) c= V by A3;
    end;
    hence V in the topology of TopSpaceNorm X by PCOMPS_1:def 4;
    hence V is open;
  end;
  now
    assume
A4: V is open;
    hereby
      let x be Point of X such that
A5:   x in V;
      reconsider z=x as Element of MetricSpaceNorm X;
      V in the topology of TopSpaceNorm X by A4;
      then consider r be Real such that
A6:   r > 0 & Ball(z,r) c= V by A5,PCOMPS_1:def 4;
      take r;
      ex t be Point of X st t = z & Ball(z,r) = {y where y is Point of X:
      ||.t-y.|| < r} by Th2;
      hence r > 0 & {y where y is Point of X: ||.x-y.|| < r} c= V by A6;
    end;
  end;
  hence thesis by A1;
end;
