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
  A is finite-ind & B is finite-ind & TM| (A\/B) is second-countable
    implies A\/B is finite-ind & ind(A\/B) <= ind A+ind B+1 by Lm4;
