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 H st TM|H is second-countable holds H is
  finite-ind & ind H<=n iff ex Bas be Basis of TM st
  for A st A in Bas holds H/\Fr A is finite-ind & ind(H/\Fr A)<=n-1
proof
  set cTM=[#]TM;
  set TOP=the topology of TM;
  let M be Subset of TM such that
A1: TM|M is second-countable;
  hereby
    defpred P[object,object] means
    for p be Point of TM,A be Subset of TM st$1=[p,A] holds$2 in TOP & (not p
in A implies $2={}TM) & (p in A implies ex W be open Subset of TM st W=$2 & p
    in W & W c=A & M/\Fr W is finite-ind & ind(M/\Fr W)<=n-1);
    assume
A2: M is finite-ind & ind M<=n;
A3: for x be object st x in [:cTM,TOP:]ex y be object st P[x,y]
    proof
      let x be object such that
A4:   x in [:cTM,TOP:];
      consider p9,A9 be object such that
A5:   p9 in cTM and
A6:   A9 in TOP and
A7:   x=[p9,A9] by A4,ZFMISC_1:def 2;
      reconsider A9 as open Subset of TM by A6,PRE_TOPC:def 2;
      reconsider p9 as Point of TM by A5;
      per cases;
      suppose
A8:     not p9 in A9;
        take{}TM;
        let p be Point of TM,A be Subset of TM such that
A9:     x=[p,A];
        p=p9 by A7,A9,XTUPLE_0:1;
        hence thesis by A7,A8,A9,PRE_TOPC:def 2,XTUPLE_0:1;
      end;
      suppose p9 in A9;
        then consider W be open Subset of TM such that
A10:    p9 in W & W c=A9 & M/\Fr W is finite-ind & ind(M/\Fr W)<=n-
        1 by A1,A2,Th13;
        take W;
        let p be Point of TM,A be Subset of TM such that
A11:    x=[p,A];
        p=p9 & A=A9 by A7,A11,XTUPLE_0:1;
        hence thesis by A10,PRE_TOPC:def 2;
      end;
    end;
    consider f be Function such that
A12: dom f=[:cTM,TOP:] and
A13: for x be object st x in [:cTM,TOP:] holds P[x,f.x]
from CLASSES1:sch 1(A3);
A14: rng f c=TOP
    proof
      let y be object;
      assume y in rng f;
      then consider x be object such that
A15:  x in dom f and
A16:  f.x=y by FUNCT_1:def 3;
      ex p,A be object st p in cTM & A in TOP & x=[p,A]
by A12,A15,ZFMISC_1:def 2;
      hence thesis by A12,A13,A15,A16;
    end;
    then reconsider RNG=rng f as Subset-Family of TM by XBOOLE_1:1;
    now
      let A be Subset of TM such that
A17:  A is open;
A18:  A in TOP by A17,PRE_TOPC:def 2;
      let p be Point of TM such that
A19:  p in A;
A20:  [p,A] in [:cTM,TOP:] by A18,A19,ZFMISC_1:87;
      then consider W be open Subset of TM such that
A21:  W=f.[p,A] & p in W & W c=A and
      M/\Fr W is finite-ind and
      ind(M/\Fr W)<=n-1 by A13,A19;
      reconsider W as Subset of TM;
      take W;
      thus W in RNG & p in W & W c=A by A12,A20,A21,FUNCT_1:def 3;
    end;
    then reconsider RNG as Basis of TM by A14,YELLOW_9:32;
    take RNG;
    let B be Subset of TM;
    assume B in RNG;
    then consider x be object such that
A22: x in dom f and
A23: f.x=B by FUNCT_1:def 3;
    consider p,A be object such that
A24: p in cTM and
A25: A in TOP and
A26: x=[p,A] by A12,A22,ZFMISC_1:def 2;
    reconsider A as open Subset of TM by A25,PRE_TOPC:def 2;
    per cases;
    suppose p in A;
      then ex W be open Subset of TM st W=f.[p,A] & p in W & W c=A & M/\Fr W is
      finite-ind & ind(M/\Fr W)<=n-1 by A12,A13,A22,A26;
      hence M/\Fr B is finite-ind & ind(M/\Fr B)<=n-1 by A23,A26;
    end;
    suppose not p in A;
      then B={}TM by A12,A13,A22,A23,A24,A26;
      then
A27:  Fr B={} by TOPGEN_1:14;
      0-1<=n-1 by XREAL_1:9;
      hence M/\Fr B is finite-ind & ind(M/\Fr B)<=n-1 by A27,TOPDIM_1:6;
    end;
  end;
  given B be Basis of TM such that
A28: for A be Subset of TM st A in B holds M/\Fr A is finite-ind &
  ind(M/\Fr A)<=n-1;
  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
  proof
    let p be Point of TM,U be open Subset of TM such that
A29: p in U;
    consider a be Subset of TM such that
A30: a in B and
A31: p in a & a c=U by A29,YELLOW_9:31;
    B c=TOP by TOPS_2:64;
    then reconsider a as open Subset of TM by A30,PRE_TOPC:def 2;
    take a;
    thus thesis by A28,A30,A31;
  end;
  hence thesis by A1,Th13;
end;
