reserve n for Nat,
        X for set,
        Fx,Gx for Subset-Family of X;
reserve TM for metrizable TopSpace,
        TM1,TM2 for finite-ind second-countable metrizable TopSpace,
        A,B,L,H for Subset of TM,
        U,W for open Subset of TM,
        p for Point of TM,

        F,G for finite Subset-Family of TM,
        I for Integer;

theorem
  for TM st TM is second-countable &
    ex F st F is closed & F is Cover of TM & F is countable &
       F is finite-ind & ind F <= n
  holds TM is finite-ind & ind TM <= n by Lm3;
