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
  ind G = -1 & G is finite-ind iff G c= {{}T}
proof
A1: {{}T}=(Seq_of_ind T).0 by Def1;
  0=-1+1;
  hence ind G=-1 & G is finite-ind implies G c={{}T} by A1,Def6;
  assume
A2: G c={{}T};
  then
A3: G is finite-ind by A1;
  then
A4: -1<=ind G by Def6;
  0=-1+1;
  then ind G<=-1 by A1,A2,A3,Def6;
  hence thesis by A1,A2,A4,XXREAL_0:1;
end;
