reserve i for Nat;
reserve R for Relation;
reserve A for set;
reserve PT for non empty TopSpace;
reserve PM for MetrSpace;
reserve FX,GX,HX for Subset-Family of PT;
reserve Y,V,W for Subset of PT;

theorem
  for f being Function of [:the carrier of PT,the carrier of PT:],REAL
st f is_metric_of ( the carrier of PT) holds PM = SpaceMetr(the carrier of PT,f
  ) implies (FX is Subset-Family of PT iff FX is Subset-Family of PM) by Th4;
