reserve r for Real;

theorem Th15:
  for M being MetrStruct, p being Element of M holds Sphere(p,r)
  c= cl_Ball(p,r)
proof
  let M be MetrStruct,p be Element of M;
  per cases;
  suppose
A1:  M is non empty;
    now
      let x be Element of M;
      assume x in Sphere(p,r);
      then dist(p,x)=r by Th13;
      then x in {q where q is Element of M:dist(p,q)<=r};
      hence x in cl_Ball(p,r) by Def15,A1;
    end;
    hence thesis by SUBSET_1:2;
  end;
  suppose
A2: M is empty;
    then Sphere(p,r) is empty;
    hence thesis by A2,Def15;
  end;
end;
