reserve T,T1,T2 for TopSpace,
  A,B for Subset of T,
  F for Subset of T|A,
  G,G1, G2 for Subset-Family of T,
  U,W for open Subset of T|A,
  p for Point of T|A,
  n for Nat,
  I for Integer;
reserve Af for finite-ind Subset of T,
  Tf for finite-ind TopSpace;

theorem Th11:
  G is finite-ind & ind G <= I iff -1 <= I & for A st A in G holds
  A is finite-ind & ind A <= I
proof
  hereby
    assume that
A1: G is finite-ind and
A2: ind G<=I;
    ind G>=-1 by A1,Def6;
    then ind G+1>=-1+1 by XREAL_1:6;
    then ind G+1 in NAT by INT_1:3;
    then reconsider iG=ind G+1 as Nat;
A3: G c=(Seq_of_ind T).iG by A1,Def6;
    -1<=ind G by A1,Def6;
    hence -1<=I by A2,XXREAL_0:2;
    let A such that
A4: A in G;
    thus A is finite-ind by A1,A4;
    then ind A<=iG-1 by A3,A4,Th7;
    hence ind A<=I by A2,XXREAL_0:2;
  end;
  assume that
A5: -1<=I and
A6: for A st A in G holds A is finite-ind & ind A<=I;
  -1+1<=I+1 by A5,XREAL_1:6;
  then reconsider I1=I+1 as Element of NAT by INT_1:3;
A7: G c=(Seq_of_ind T).I1
  proof
    let x be object such that
A8: x in G;
    reconsider A=x as Subset of T by A8;
A9: I=I1-1;
    A is finite-ind & ind A<=I by A6,A8;
    hence thesis by A9,Th7;
  end;
  then G is finite-ind;
  hence thesis by A5,A7,Def6;
end;
