reserve r for Real;

theorem Th13:
  for M being MetrStruct, p, x being Element of M holds x in
  Sphere(p,r) iff M is non empty & dist(p,x) = r
proof
  let M be MetrStruct,p,x be Element of M;
  hereby
    assume
A1: x in Sphere(p,r);
    then reconsider M9 = M as non empty MetrStruct;
    reconsider p9 = p as Element of M9;
    x in {q where q is Element of M9:dist(p9,q)=r} by A1,Def16;
    then ex q be Element of M st x=q & dist(p,q)=r;
    hence M is non empty & dist(p,x)=r by A1;
  end;
  assume M is non empty;
  then reconsider M9 = M as non empty MetrStruct;
  reconsider p9 = p as Element of M9;
  assume dist(p,x)=r;
  then x in {q where q is Element of M9:dist(p9,q)=r};
  hence thesis by Def16;
end;
