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
  ind TM2 = 0 implies ind [:TM1,TM2:] = ind TM1
proof
  assume
A1: ind TM2=0;
then A2: TM2 is non empty by TOPDIM_1:6;
  then
A3: ind[:TM1,TM2:]<=ind TM1+0 by A1,Lm5;
  set x=the Point of TM2;
  reconsider X={x} as Subset of TM2 by A2,ZFMISC_1:31;
  per cases;
  suppose
A4: TM1 is empty;
    then ind TM1=-1 by TOPDIM_1:6;
    hence thesis by A4,TOPDIM_1:6;
  end;
  suppose
 TM1 is non empty;
    then ind[:TM2|X,TM1:]=ind TM1 by A2,BORSUK_3:13,TOPDIM_1:25;
    then
A5: ind[:TM1,TM2|X:]=ind TM1 by TOPDIM_1:28;
A6: [:TM1,TM2|X:] is SubSpace of[:TM1,TM2:] by BORSUK_3:15;
    then [#][:TM1,TM2|X:]c=[#][:TM1,TM2:] by PRE_TOPC:def 4;
    then reconsider cT12=[#][:TM1,TM2|X:] as Subset of[:TM1,TM2:];
    [:TM1,TM2|X:]=[:TM1,TM2:]|cT12 by A6,PRE_TOPC:def 5;
    then ind cT12=ind TM1 by A5,TOPDIM_1:17;
    then ind TM1<=ind[:TM1,TM2:] by TOPDIM_1:19;
    hence thesis by A3,XXREAL_0:1;
  end;
end;
