
theorem
  for M being non empty MetrSpace,Sm being sequence of M, St being
  sequence of TopSpaceMetr(M) st Sm=St holds Sm is convergent iff St is
  convergent
proof
  let M be non empty MetrSpace,Sm be sequence of M, St be sequence of
  TopSpaceMetr(M);
  assume
A1: Sm=St;
  thus Sm is convergent implies St is convergent
  proof
    assume Sm is convergent;
    then consider x being Point of M such that
A2: Sm is_convergent_in_metrspace_to x by METRIC_6:10;
    reconsider x9=x as Point of TopSpaceMetr(M) by TOPMETR:12;
    St is_convergent_to x9 by A1,A2,Th28;
    hence thesis;
  end;
  assume St is convergent;
  then consider x9 being Point of TopSpaceMetr(M) such that
A3: St is_convergent_to x9;
  reconsider x=x9 as Point of M by TOPMETR:12;
  Sm is_convergent_in_metrspace_to x by A1,A3,Th28;
  hence thesis by METRIC_6:9;
end;
