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 Th13:
  for H st TM|H is second-countable holds H is
  finite-ind & ind H<=n iff for p,U st p in U
  ex W st p in W & W c=U & H/\Fr W is finite-ind & ind(H/\Fr W)<=n-1
proof
  let M be Subset of TM such that
A1: TM|M is second-countable;
  hereby
    assume that
A2: M is finite-ind and
A3: ind M<=n;
    let p be Point of TM,U be open Subset of TM such that
A4: p in U;
    reconsider P={p} as Subset of TM by A4,ZFMISC_1:31;
    TM is non empty & not p in U` by A4,XBOOLE_0:def 5;
    then consider L be Subset of TM such that
A5: L separates P,U` and
A6: ind(L/\M)<=n-1 by A1,A2,A3,Th11,ZFMISC_1:50;
    consider W,W1 be open Subset of TM such that
A7: P c=W and
A8: U`c=W1 and
A9: W misses W1 and
A10: L=(W\/W1)` by A5,METRIZTS:def 3;
    take W;
    W c=W1` & W1`c=U`` by A8,A9,SUBSET_1:12,23;
    hence p in W & W c=U by A7,ZFMISC_1:31;
    Cl W misses W1 by A9,TSEP_1:36;
    then Cl W\W1=Cl W by XBOOLE_1:83;
    then Fr W=(Cl W\W1)\W by TOPS_1:42
      .=Cl W\(W\/W1) by XBOOLE_1:41
      .=Cl W/\L by A10,SUBSET_1:13;
    then Fr W c=L by XBOOLE_1:17;
    then
A11: M/\Fr W c=M/\L by XBOOLE_1:27;
    M/\L c=M by XBOOLE_1:17;
    then
A12: M/\L is finite-ind by A2,TOPDIM_1:19;
    then ind(M/\Fr W)<=ind(M/\L) by A11,TOPDIM_1:19;
    hence M/\Fr W is finite-ind & ind(M/\Fr W)<=n-1 by A6,A11,A12,TOPDIM_1:19
,XXREAL_0:2;
  end;
  set Tm=TM|M;
  assume
A13: for p be Point of TM,U be open Subset of TM st p in U ex W be
open Subset of TM st p in W & W c=U & M/\Fr W is finite-ind &
ind(M/\Fr W)<=n-1;
A14: for p be Point of Tm,U be open Subset of Tm st p in U ex W be open
  Subset of Tm st p in W & W c=U & Fr W is finite-ind & ind Fr W<=n-1
  proof
A15: [#]Tm=M by PRE_TOPC:def 5;
    let p be Point of Tm,U be open Subset of Tm such that
A16: p in U;
    p in M by A15,A16;
    then reconsider p9=p as Point of TM;
    consider U9 be Subset of TM such that
A17: U9 is open and
A18: U=U9/\the carrier of Tm by TSP_1:def 1;
    p9 in U9 by A16,A18,XBOOLE_0:def 4;
    then consider W9 be open Subset of TM such that
A19: p9 in W9 & W9 c=U9 and
A20: M/\Fr W9 is finite-ind and
A21: ind(M/\Fr W9)<=n-1 by A13,A17;
    reconsider W=W9/\the carrier of Tm as Subset of Tm by XBOOLE_1:17;
    reconsider W as open Subset of Tm by TSP_1:def 1;
    take W;
    thus p in W & W c=U by A16,A18,A19,XBOOLE_0:def 4,XBOOLE_1:26;
    reconsider FrW=Fr W as Subset of TM by A15,XBOOLE_1:1;
A22: FrW c=Fr W9/\M by A15,TOPDIM_1:1;
    then
A23: FrW is finite-ind by A20,TOPDIM_1:19;
    ind FrW<=ind(Fr W9/\M) by A20,A22,TOPDIM_1:19;
    then ind Fr W<=ind(Fr W9/\M) by A23,TOPDIM_1:21;
    hence thesis by A21,A23,TOPDIM_1:21,XXREAL_0:2;
  end;
  then
A24: Tm is finite-ind by TOPDIM_1:15;
  then
A25: M is finite-ind by TOPDIM_1:18;
  ind Tm<=n by A14,A24,TOPDIM_1:16;
  hence thesis by A25,TOPDIM_1:17;
end;
