reserve PM for MetrStruct;
reserve x,y for Element of PM;
reserve r,p,q,s,t for Real;
reserve T for TopSpace;
reserve A for Subset of T;
reserve T for non empty TopSpace;
reserve x for Point of T;
reserve Z,X,V,W,Y,Q for Subset of T;
reserve FX for Subset-Family of T;
reserve a for set;
reserve x,y for Point of T;
reserve A,B for Subset of T;
reserve FX,GX for Subset-Family of T;
reserve x,y,z for Element of PM;
reserve V,W,Y for Subset of PM;

theorem
  for D be non empty set, f be Function of [:D,D:],REAL st f
  is_metric_of D holds SpaceMetr(D,f) is non empty
proof
  let D be non empty set, f be Function of [:D,D:], REAL;
  assume f is_metric_of D;
  then SpaceMetr(D,f) = MetrStruct(#D,f#) by Def7;
  hence thesis;
end;
