 reserve X for RealUnitarySpace;
 reserve x, y, y1, y2 for Point of X;

theorem
for S be RealUnitarySpace
holds
  TopSpaceNorm RUSp2RNSp S = TopUnitSpace S
proof
  let S be RealUnitarySpace;
  set PM = MetricSpaceNorm RUSp2RNSp S;
  set FPM = Family_open_set PM;
for M being Subset of S holds
 M in FPM iff for x being Point of S st x in M holds
ex r being Real st r > 0 & Ball (x,r) c= M
proof
let M be Subset of S;
reconsider N = M as Subset of PM;
hereby assume A1:M in FPM;
 thus for x being Point of S st x in M holds
ex r being Real st r > 0 & Ball (x,r) c= M
proof
  let x be Point of S;
  assume A2: x in M;
  reconsider y=x as Element of PM;
consider r being Real such that
A3: r > 0 & Ball (y,r) c= N by A2,A1,PCOMPS_1:def 4;
take r;
thus 0 < r by A3;
thus Ball (x,r) c= M by A3,Th13;
end;
end;
assume A4:for x being Point of S st x in M holds
ex r being Real st r > 0 & Ball (x,r) c= M;
for y being Element of PM st y in N holds
ex r being Real st
 r > 0 & Ball (y,r) c= N
proof
  let y be Element of PM;
  assume A5: y in N;
  reconsider x=y as Point of S;
  consider r being Real such that
 A6: r > 0 & Ball (x,r) c= M by A4,A5;
take r;
thus 0 < r by A6;
thus thesis by A6,Th13;
end;
hence M in FPM by PCOMPS_1:def 4;
end;
hence thesis by RUSUB_5:def 5;
end;
