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 Th4:
  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 the carrier of PM = the carrier of PT
proof
  let f be Function of [:the carrier of PT,the carrier of PT:],REAL;
  assume
A1: f is_metric_of the carrier of PT;
  assume PM = SpaceMetr(the carrier of PT,f);
  then PM = MetrStruct(#the carrier of PT,f#) by A1,PCOMPS_1:def 7;
  hence thesis;
end;
